Welcome sirs and gals to my Amumu guide. Its a rather different build from most other jungle guides but it serves its purpose well. Of which is to jungle, gank and dominate the other team.
So without further ado, we shall begin!

I find that the Greater Quintessence of Magic Penetration, Greater Mark of Magic Penetration and Greater Glyph of Magic Penetration have good synergy with Amumu's passive Cursed Touch. The Greater Seal of Health compliment Veteran's Scars .

Smite for the jungle and Flash for the people that use Flash, so that you can Flash on them after they Flashed away... FLASH. There is no escape. It is also very practical to use as an escape mechanism after engaging the enemy team with Bandage Toss and Curse of the Sad Mummy.

0-10-20, why? I get the strict minimum in the defensive tree, just enough to get Veteran's Scars and Bladed Armor . I shove the rest in the support tree for Wealth , Transmutation and Intelligence . The Runic Affinity mastery makes it all a bit easier. I like the feel of it.
This makes Amumu's first clear of the jungle very easy, and this first clear of the jungle is fundamental towards his success.

Cursed Touch - Passive - With Abyssal Mask you can lower you're target by up to 55 MR.
Bandage Toss - Very useful for ganking, initiating and catching escapees.
Despair - Unique spell that is extremely effective against high HP opponents, such as;

Ancient Golem
Lizard Elder
Dragon
Baron Nashor
Champion Tanks

Tantrum - AOE spell that will eventually decimate waves of minions. Its passive greatly helps you jungle in the early game.
Curse of the Sad Mummy - A very powerful Ult that has a major impact in team fights. Don't be shy to use it as often as possible.

Oooh whats going on?!
The first 7 items, Cloth Armor and Health Potion x6 (or 5 if you changed the Masteries) are the basic for the way of the jungle. You will sell the Cloth Armor mid-late game for 210g losing 90g, which is insignificant.
I get my Boots next with ~2 Health Potion to get around the map quicker.
Make you're way towards building a Hextech Revolver and get Health Potion as needed.
Complete Will of the Ancients, you're first core item.
Finish you're Mercury's Treads, get Chain Vest and Negatron Cloak. I usually get Glacial Shroud next for the mana and CDR. After comes Giant's Belt for HP. Then finally my beloved Rylai's Crystal Scepter for slow and more DMG. Finish Abyssal Mask for more hurting and then Frozen Heart for more CDR and armor.

You have one item slot left and the game isn't over?! I usually get a Needlessly Large Rod to end it. If you get there, finish with a Rabadon's Deathcap.

The jungle route for Amumu is typical; Ancient Golem> Wolves> Wraiths> Lizard Elder> Golems. You won't be quite LVL 4 yet and therefore you will not have Q either after your first clear of your side of the jungle. So after clearing the Golems I usually recall and get my Boots, then I go to the Wolves and try to gank top/midle or move to the Wraiths>Golems and gank bot. (You could get Q at level 3 if you see that bottom has an easy gank situation, otherwise at that LVL its a waste of effort, get your Boots first.)

Once you complete Will of the Ancients (if all goes well around 10 minutes in game) you can solo Dragon and devastate the land of the jungle without any worries (Bring a Health Potion or 2 to solo Dragon).
After Will of the Ancients, farming is easy. All minions are your pray without resistance. Not to mention you should be racking kills from ganks. You should be well off on gold.

Few, you are now master of the jungle, your ganks were brutal and now some turrets are GONE.
Now that that's all done the 5v5 team fights will be more frequent. And so the tanky items roll out and we move on to Team Work.

Amumu's job as a jungler is to gank, which isn't very hard to do considering that you land Bandage Toss. And even then you'll maybe miss more often then not (Hopefully not too often). You want to try to get as close as possible to the enemy before tossing, unless you are almost certain to land it. So while ganking don't try to always Toss from the Bush. Try to run behind the enemy and narrow his exits giving you less chance to miss.

This build is different, it thinks outside of the box and it wants you to have fun.
This also works for other AP/Tank Champions such as Alistar or Nunu & Willump.

This is a viable build for Ranked games as long as you have a Support-Tank in bot lane or Tank/Off-Tank in top lane.
Amumu is vulnerable to bans because of his ULT.

I hope that you enjoy this build as much as I do :)