This is my Amumu build (no kidding?!), I haven't played Amumu in a good while, but that doesn't mean that i don't know what might work for him (hence, Theory Amumu)
I haven't tested the build YET! but I will soon!

The idea of this build, is to get enough MP/5 to have Despair active at ALL times (40 MP/5 is needed) and also have enough for the rest of your skills (so about 50 MP/5). then, after you get enough, proceed building AP and defensive items. I'll get more into detail about that later.
ENJOY!

+ Great damage from Despair, due to the AP
+ Have it on at all times
+ Pretty tanky, although not as much as a full-tank
+ good Mana pool
+ Decent M.Speed

- As soon as the enemy notices your DPS, they will (probably) target you
- Too balanced to be awesome at any One thing (not main DPS, not main Tank)
- A bit costly, but not stupidly over-expensive

There are most likely more of both, but i can't think of them yet.

I go for 0/21/9. Because this build lacks a bit of the tank-y nature, and the Utility will help your on-going-ness

I am personally not the best at Masteries, so they are subject to change, along with the Runes (That I don't even have a section for, because I know very little about it)

Chalice of Harmony - A great item for your mana pool!, this item will help you a LOT until late game (and sometimes there too). It is one of my favourite items, and fits this build just great

Tear of the Goddess - Great item, gives a ton of mana at the end, and gets built into Archangel's Staff

Boots of Swiftness - I choose these boots, because i feel at Amumu is Slooooooo~oooow, and therefore needs speed. You can also choose Sorcerer's Shoes for some extra punch or Mercury's Treads if the enemy is full of CC and/or has a lot of casters.

Archangel's Staff - Lots of MP/5, lots of mana, lots of AP. Need I say more?

Deathfire Grasp - it gives the very last needed MP/5, and it gives some needed AP. Also, if you're good at using Items' actives, this is absolute GOLD!

---OPTIONALS---

Rod of Ages - it gives more HP and mana, which is always welcome, and it also gives that amazing AP that we need!

Rylai's Crystal Scepter - HP, AP, Slows! i love this item, it just works so well when chasing!

Rabadon's Deathcap - it is the ULTIMATE in AP! seriously, you can't not get it if you're going for AP

Zhonya's Hourglass - it gives you some needed AP and armor, plus it has a great active that might just save you.

Thornmail - if the enemy team is full of melee's and you want to be a tank, you need this. it's great.

Sunfire Aegis - Everyone loves this item. I'm not that big a fan, usually, but if you add this with your Despair it'll be just great. It also gives some HP, which is great.

I personally take
Flash
and
Ghost

for all tanks or tank-ish champs, because you'll want to get into and out of the battle quickly.

Other choices include:

Clarity - Due to the massive MP/5 of this build, it'll only help at start. but at start it WILL help a lot.

Heal - I'm a fan of Heal, it rocks. It's a good choice on this build, due to it's squishyness until later in the game.

Ignite - just because i love ignite. it has so many uses. all of which include destruction >:3

Exhaust - it can at times turn a battle, due to the damage reduction of it. But you can also chase with it.

I might add more to this build, but this is what it looks like now!
please leave critique and comments! I'd love to hear what you think.