Standard jungle amumu build with a balanced item order throughout the game. One problem i've always found with laning amumu is that you always have trouble farming. Using tantrum to last hit is a massive waste of mana unless you time every one perfectly and that doesnt always work out. Early game you will have to choose between using your mana to farm the best you can or to counter harrassment. I find you cant do both successfully resulting in you having to leave lane too often to regen health.

My play style;
This is pretty simple. Early game is just to farm. Jungle, support your lanes and gather creep xp, go for ganks. This is easier said than done. Amumus damage output is poor early game. Depending on whos on the lane you want to gank with pay attention to what stuns, slows, and damage output your team has, compared to how aggressive your enemy is playing, who you will be ganking, what summoner abilities they may have to escape. Its all important in grabbing first blood or an early assist. As i explain in more detail later, if an enemy is very aggressive on a lane, to the point where they may tower dive at level 6, use this to your advantage, bandage toss from behind a bush, ulti + fortify. Mid game (around level 10-12 or when you have build up to your first chain vest) you should be out of the jungle and helping your team take towers and with it map control. Don't worry, aslong as you dont do anything careless, you can tank well at this stage. Coming into late game you have to be smart. Do not tank towers in a 5v5 fight until your randuins is complete and then you should only do it if you really need to. I find the fights easier when its in neutral areas where you dont have to worry about your damage intake and you can focus on your role. Always go for the DPS that are targetting your own DPS. Your damage output at this stage is fairly decent. Any tanks near you will be taking lots of damage from your despair. If the other team initiates, run right up to the main dps and bandage toss him. The short stun can turn the fight for your team. This works better on melee DPS as they can be focused easier than a ranged who may be hiding behind the tanks. Make sure your team focuses this guy then go for any ranged behind the tanks. At any stage where you have enough enemies in your range use your ulti and take that opportunity to nuke the biggest threat. i.e if you have a tank, an offtank and a melee dps stunned in your ulti your team should ignore them and focus the ranged, e.g Ashe, Trist, Kogmaw. If these can be dropped in the first 5 seconds of a team fight then the fight should be yours. Just remember to use your bandage toss to disable anyone attacking your DPS.

Jungle rotation;
You will need the mana buff to complete the first circuit. Make sure to wait in the bush near mid river before minions spawn incase a team decides to go for first blood before the game has really even begun. Despair works really well on golem at level 1 but its useless if your being ganked. I will warn you teams have tried to gank me in about 80% of my games jungling amumu. They have never succeeded.
On your first circuit if you havnt lost any health because of a gank you can easily go ancient golems, wolf camp, wraith camp, lizard camp, double golems and blue pill with 2 bars and just a bit of mana left to buy your first item.
I always like to have a decent amount of health when roaming the jungle on my first circuit as you can be caught out if your not careful. Make sure to pop your first health potion once your first bar of health is gone in the ancient golem fight. Then i pop a potion between each camp. (i.e one during golem, one after golem, one after wolves, one after wraiths, one during lizard.) This reduces the chance of you being caught out mid jungle with not enough hp to escape, the only risky camp is the last double golems and thats close enough to a tower to be ok.

The reason i have chosen this build sequence;
Obviously taking the armour and health pots for jungling is always the same. Starting at blue is a must for amumu. After your first rotation you will have enough gold to buy a ruby crystal (475g). You will need to go back to regen health anyway so make sure you grab it. Wolf camp should have respawned by the time you get there so go through the camps again and return when you have enough to finish your heart of gold and boots of speed (550g). At any point you need to take over a lane or just grab abit of farm do so. It wont affect your jungle at all. Grab buffs on respawn and try to do a full rotation each time. Remember you have fortify if you need it. On your next port home you should grab a sapphire crystal and complete your boots, if alls gone well complete your catalyst too. Try to grab your first negatron cloak before tanking any real fights. Sometimes you won't have the choice but your team shouldnt be in a position where they need you in the fights before you have it. Once you do your already surprisingly tanky. After completeing banshees veil we come to my favourite item on amumu. The glacial shroud (which later becomes your frozen heart). Despite the nerf in patch v1.0.0.111 to the CDR (reduced by 5%), the combine cost was reduced to match. This item is so overlooked. The armour, mana, and CDR it gives are massive for amumu, and keeping despair on in longteam fights can be a strain on your mana without bluebuff or a decent mana pool. I personally find it suits your carry or support alot more if you can build to not need blue buff after the early game phase is finished. Also with the strength of spirit mastery maxed the more mana you have, the more health regen per 5 seconds you get. Your next item, force of nature provides even more MR and health regen. After completing this you just work on finishing your frozen heart and randuins omen. Randuins gives you more health, more armour, more health regen, even more CDR, and some nice passive and active slows to enemy movement speed and attack speed. By the time this is completed you should be unstoppable.
I have only ever got to this stage in my build and i've never needed another item, but items I believe could be useful would be;

Guardian angel - Just because you can.
Sunfire cape - It may be late in the game for this one but with the extra hp and armour, if an enemy team is tower hugging and you wan to finish, it really adds to your ability to tank a tower during a team fight.
Abyssal sceptre - For that little bit extra damage with some nice extra magic resist.
Rylai's crystal sceptre - Extra hp, a little AP, and a constant 15% slow to any enemy in range of your despair. No-one will get away.

The reason i have chosen to spec into fortify and select it as a summoner spell is pretty simple. Its becoming more popular for tanks to take fortify just as a support champion would take clairvoyance. First of all when fortify is off cooldown it gives you that extra attack damage in early game jungling. This is useful with the attack speed runes in your first part of jungling to get you levelling quickly. The time i find fortify really comes into its own is in those games when things arent going your teams way. Its quite often now for teams to dive towers 5v2 or 5v3. The number of times one of our low hp carries will bait a team into tower range for us to turn a 3v5 fight into an ace is laughable. The mixture of a well timed ulti + fortify (with splash damage due to the mastery spec)can rip a team apart before they realise they're still in tower range.

I use this build in all of my games but I find it really suits a well co-ordinated team, which is why I recommend it be used in premade only. Please dont take this build into a solo queue and lose because your team dont respond to what you plan to do, only to downvote this build.

Here are my scores from my last few games with amumu;
http://img534.imageshack.us/i/lolscores.png/

I hope you all find this build as fun to play as I do.