Perseverance is being removed, so this guide will be overhauled in the near future.

The goal of this guide is to adapt Amumu perfectly to the new jungle. First, let's look at the problems introduced by the new jungle.

Problems:
1. Lower experience gain from the jungle.
2. Less gold gained from the jungle.
3. Due to low experience gain, ganking becomes more difficult to incorporate into your playstyle since it is easy to fall behind in level if you are trying to gank too often.
4. Minion stealing will probably become more common on the first run.
5. There are many, many more...

Basically, I decided junglers, especially Amumu, need a few essential things to be successful in the new jungle.
1. Speed: The ability to move from creep to creep as quickly as possible increases our "experience generation rate", if you will. Also, ganking is easier if you are fast.
2. Health and Mana Regeneration: The ability to stay in the jungle longer also increases our "experience generation rate". Also, you are more able to gank on impulse if you constantly have sufficient health and mana.

With this in mind, I came up with several specific goals:
1. Level to 6 as fast as possible.
2. Obtain high health and mana regeneration as early as possible.
3. Maximize gold/second rate early.
4. Obtain boots as early as possible.

With these specific goals in mind, I devised a build that achieves all of these goals.

Runes:
Greater Mark of Magic Penetration There are no changes here. Nice damage boost to all of Amumu's skills. These are the only marks Amumu should ever have.

Greater Seal of Armor There are no changes here, either. Fantastic damage reduction from minions and AD Champions.

Greater Glyph of Scaling Magic Resist Again, no changes here. These glyphs are common for tank builds. Provides a nice magic resist boost throughout the game.

Greater Quintessence of Vigor This is where things start to change. These quintessences are vital to Amumu's first jungle run. Let's do a simple calculation. lets say you start fighting minions at 1:40, and finish your run at 4:00 (This is what happens in this guide, btw). Over this time, three of these quintessences will have healed you for a total of 227 health. Three flat health quintessences provide 78 health. This 149 health difference is the difference between living and dying on the first jungle run.

Summoner Skills:
Smite 100% necessary for Amumu to jungle
Flash Able to be used as an escape or offensively. Even with the recent range reduction and cooldown increase, it is still the best summoner skill for Amumu.

First, I'd like to catagorize the masteries by their purpose.

Surviving and Sustaining the Jungle:
Hardiness Reduces damage from minions and AD champions.
Tough Skin Again, reduces damage from minions.
Vigor Provides 84 health during the first jungle run. Priceless.
Bladed Armor Allows slightly faster elimination of minion camps, which in turn reduces the number of times they can attack you, thus lowering the damage taken from minions during jungling. Also, increases "experience regeneration rate" by speeding up the jungle process.
Expanded Mind Not as important as the mana regeneration benefits it allows you to obtain.
Meditation Provides just enough mana regeneration to carry you through the second jungle run.
Swiftness Little boost to speed increases "experience generation rate".
Runic Affinity Absolutlely necessary for Amumu to hold blue as long as possible.
Perseverance A 9% boost to both health and mana regeneration is key for Amumu to sustain his jungle.

Gold Generation:
Summoner's Resolve 10g reward for using Smite. Nets 20g on first jungle run. Nice boost throughout the game.
Greed 2g/10sec. Nets 30g on first jungle run. Again, nice boost throughout the game.
Wealth 40 extra starting gold. I like to think of this as a net of 40g on the first jungle run.

Note on Mastery Choices:
Most builds used with Amumu have 21 points in the defensive tree and 9 in the utility tree. I deviated from this path to obtain the necessary gold to obtain Amumu's starting items quickly. However, I feel the utility tree provides better benefits to Amumu's ability to jungle than the defensive tree. Benefits include improved health and mana regeneration, improved g/sec rate, improved speed, and improved buff duration. For me, the sacrifice of tankiness is worth the boost to my jungle.

Ok, so as we all know most junglers start with a cloth armor and 5 health potions. Health potions cost 35g each, so 5 of them would cost 175g. Quite a waste if you ask me, especially if you can avoid buying them. The good news is, Amumu can jungle without health potions.

Starting Items: Regrowth Pendant

That is it. Do not buy any health potions. You will have 80g left over unused to start with.

After completing the first jungle run, with all the correct runes and masteries, you will be able to buy a Philosopher's Stone and Boots at 4:20. You should be back at base by 4:10, so you may have to wait a little before going back out into the jungle. The benefit these items give you are simply awesome. They allow you to sustain your jungle, while giving you an additional 5g/sec and a movement speed increase of 315 to 386. All of this while sacrificing as little time as possible.

For fun, let us calculate the gold gained up until the 4:20 mark with and without masteries.

Clearing the minions in the jungle route will net you 345g. Without masteries, you would have 475g + 345g + 1.4g/sec * 170sec = 1058g at the 4:20 mark, leaving you 88g short of buying both Philosopher's Stone and Boots. With masteries we will have gained 475g + 40g + 345g + 1.6g/sec * 170sec = 1152g, which is just greater than the 800g + 350g = 1150g needed to buy Philosopher's Stone and Boots.

Run 1:

Start the game by buying a Regrowth Pendant and putting a point into Tantrum, then head to wolves. If you can, try to get your teammates to protect the other camps from counter-junglers, especially wraiths and blue.

Wolves will spawn at 1:40. After they spawn, initiate with Tantrum. Immediately start auto-attacking the large wolf while spamming Tantrum continuously until the large wolf is one hit away from death. Auto-attack one of the small wolves once, then finish the large wolf with Tantrum. One auto-attack should finish the small wolf you auto-attacked once before, while 2 will be necessary to kill the last small wolf. You should have between 300 and 400 health depending on the amount of times the wolves crit you.

Now, head to blue. Don't start fighting until you have 420 health. This may mean you will have to wait a little, but believe me when I say its worth it. Initiate with Tantrum and start to auto-attack the blue golem. Continue to auto-attack while spamming Tantrum until blue is around 500 health, in which case you should use Smite to finish it off. The lizards should already be dead. You should be level two now and close to death, but not quite dead, so head to wraiths and put a point in Despair.

Once you reach wraiths, initiate with Tantrum, flick on Despair, and start auto-attacking the blue wraith. Keep spamming Tantrum until the blue wraith is dead. Turn off Despair and auto-attack each of the small wraiths once to kill them. Now, head back to wolves.

Initiate wolves with Tantrum, flick on Despair, and start auto-attacking the large wolf. Turn off Despair once all the wolves are dead. Put a point in Tantrum and head to red golem.

Initiate red golem with Tantrum, flick on Despair, and start auto-attacking red golem. Keep spamming Tantrum the entire fight and Smite the red golem once the lizards are dead. Turn off Despair once the camp is clear. Now head to small golems.

Don't start fighting the golems until you have 200 health. Intitiate the golems with Tantrum, flick on Despair, and start auto-attacking the bigger golem. Keep spamming Tantrum throughout the fight. Once the bigger one is dead, start auto-attacking the other one. As soon as it dies, turn off Despair, put a point into Bandage Toss, and b to base. Golems should die around the 4 minute mark.

Notes For Run 1: The jungling itself is actually quite easy to do after a few tries. The key is to wait until you have enough health to fight blue and the mini golems at the end. Remember, have 420 health before fighting blue and 200 before fighting minis. Also, mana is not an issue at all throughout the first run, so always have Despair on throughout the entire fight with every camp. Be warned, it takes close to perfection to complete the first run in four minutes, so take your time if you are afraid of dying.

Run 2:

At base buy Philosopher's Stone and Boots. At 4:20, head towards wolves. Clear wolves, then wraiths, then little golems. When fighting the little golems, go ahead and Smite the larger golem. Blue will run out while fighting little golems. Repeat your run by clearing wolves, then wraiths, then golems again. Don't Smite the larger small golem this time, as you will need Smite for blue. Try to conserve mana through this run by making sure to turn off Despair once the larger minion of the group is dead. Once golems are dead, clear wolves again. After wolves are dead, head straight to blue. You should be getting there right as it respawns at around 7:20, depending on how long it took to kill the first blue. Kill the blue golem using the same method as before while finishing it off with Smite. Blue should die around 7:35. At this point you will be level 6, so put a point in Curse of the Sad Mummy and try to gank top or mid.

Notes on Run 2: If done correctly, you should find yourself waiting a couple of seconds for wolves to respawn when you get back to them the first time. This is okay. Also, you should be getting to blue before it respawns too. I really like ending on blue so the enemy jungler can't steal it and it sets you up for a glorious round of ganks.

Further Clarification: This represents the fastest and best way I've come up with to level Amumu to 6. I realize this is a very ideal jungle, meaning it assumes no one messes with you, and your team doesn't need you until almost 8 minutes into the game. I believe the first run, however, is mandatory. With Philosopher's Stone and Boots, you won't "need" to go back and buy other items for a while, which frees you up to continuously jungle and gank. After the first run, you have the option to cover a lane, gank, or jungle, depending on what your team needs.

Video of Jungle: Skip to 1:35. I only made a few little mistakes, but you should still be able to get the idea. I also apologize for the quality.
Play Video

1. Regrowth Pendant - Already explained.
2. Philosopher's Stone and Boots - Already explained.
3. Boots of Swiftness - These boots are spectacular. After the new jungle changes, I think it is imperative for Amumu to get these as soon as possible in order to jungle quickly and move quickly from lane to lane. Your movement speed will jump from 386 to 406. In fights, they allow you to position yourself better than other boots. For Amumu, positioning is everything. It's also fun to be able to chase when necessary.
4. Sunfire Aegis - This is still one of my favorite items for Amumu. The health and armor will be much needed at this point. Also, the AOE damage allows for faster jungling, not to mention more damage in team fights. At level 10, you will easily be able to solo dragon with this item.
5. Abyssal Mask or Force of Nature - I haven't decided yet which item I like better. Therefore, I will say say get Abyssal Mask if your team is winning, since it is a more offensive item than Force of Nature. However, I love Force of Nature because of how well it fits with this build. Your movement speed will be ( 310 + 90 ) * ( 1.095 ) = 433, furthering your ability to move quickly across the map. The health regeneration prevents you from having to b as much. Since you invested in Perseverance, any health regeneration is that much more helpful. Also, the 76 MR is really sweet for a tank.
6. Shurelya's Battlesong - The health, CD reduction, and health and mana regeneration alone are worth upgrading Philosopher's Stone at this point. Not to mention it's active, which can easily save teammates and serve as an initiation tool.
7. Frozen Heart - More armor, mana, and CD recuction, along with a passive that hurts the enemies AD carries. All these things Amumu benefits from.
8. Abyssal Mask or Force of Nature - There is no reason not to get both.

Warding: Whenever you b to get items, if you have left over gold BUY WARDS. Remember, wards win games. I generally like to keep two with me at all times. The two places I always keep warded are shown on the map below as circles that are half red and half blue. These wards render ganks on mid useless, and they also help track the jungler since they cover four of the eight entrances to the river. I also place wards on the red circles if I'm on the red team, and wards on the blue circles if I'm on the blue team. These wards cover the other possible entrances the enemy can access the river. For those wards, I like to ask my teammates to buy them, as it is unrealistic to maintain 4 wards at all times.

I have been jungling as Amumu for a long time now, and it has never gotten old for me. The changes to the jungle threatened the viability of a jungling Amumu to some degree, but I think my build salvages what is left the best it can.

Hopefully some of you will try this build, as I would love some feedback. Thanks for reading!