So Hi,

I use this build on my amumu on 3v3s. Haven't tested it yet on 5s so don't hate if it won't work :).
Anyway as you can see by the item build i mostly go cooldown reduction. This build has never failed me. You do the required damage, have single target stuns every 2 seconds, ult cooldown is massivly reduced and you can tank preety darn good. Read the rest, good luck in the twisted treeline and rate and comment. I DO NOT RECOMMEND THIS BUILD FOR RANKED! mainly becouse I also haven't tried it yet.

PROS
-good cc-er (crowd controller)
-aoe stun (who doesn't like that)
-practically unkillable late game (since you have 4k (or more) hp, about 200 armor and magic resist)
-even though he's tanky he does good damage


CONS
-mana problems early game (if you go defense tree masteries)
-you won't be focused in fights so you have to pay close attention to your allies so you can stun with bandage toss and use your ult in the right time (I say this becouse I don't pay much attention xD)
-frankly...other than that I can't find any more cons

For masteries i go 9/0/21. Yes this is a tank Amumu, but you are still a mana user, and since you get enough armor, magic resist and everything from runes and items I don't see why not go for some utility masteries. This build have worked for me better than any defense build I saw.

These runes are as you can see for survival. Flat health marks, well I think they are a must have for a tank. Armor seals are very good, they help a lot, but if you lack magic resist or hp/5 you can switch them for those. Health per 5 glyphs are (to me) also a must have, but yes any armor, magic resist, flat health (even attack damage :P) work as well. For quintessences i use 2 magic resist and 1 move speed (just for catching those pesky runners)

I like to max my bandage toss first just to help with those first kills. I take one tantrum at lvl 2 then bandage toss again at 3. I get despair at level 4 just becouse it does little damage early game (since it does a portion of your enemy's hp so it works better when they have more hp). Leave your despair for later levels, max Q (bandage toss) and E (tantrum) first. And ofcourse get curse of the sad mummy at levels 6, 11 and 16.

I like to start with doran's shield just for the health and armor. But your first item you get must be glacial shroud for the mana and cooldown reduction so you can harras early game. But before you get full glacial shroud you buy boots of speed. After you get Ionian boots of Lucidity you get soul shroud and right after that rush frozen heart. This item is great against attack speed champions and/or any other physical damage champ. Banshee's veil is a very usefull item for amumu, you get hp, mana and that spell negate every 45seconds. But even so you can get guardian angel insted of it but if you do I recommend you get that last then. Next if you are playing against ap champs go for force of nature. If your enemies consist more of an physical type than magic go for warmogs armor first but get force of nature after just for the hp regen.

After trying every single combination of summoner spells that are good for amumu I found that Ghost and Flash are the best. Why? Well you have escape mechanisms and/or they are good chasing spells also :)
Instead of Flash you can also get Exhaust.

Well I have a premade 3v3 team that consists of me (Amumu), Ryze and Brand. We found out that this team works GREAT! If you level your bandage toss first you're a shue in for the first blood.

Now for the laning:
We let ryze go top solo and amumu and brand go bot. Why? Becouse, well, brand is no good solo in 3s. Since you're a tank amumu you can go for dragon as soon as you hit level 5. With the damage brand and ryze put out it will be no sweat. Get dragon everytime he spawns. At level 9 if you farmed well you can kill him solo no problem. If one of your teammates is under farmed leave him 1 lane and you go clear the jungle. The green wolf buff in 3s is godlike for you but the black wraith move speed buff is also good.