I think that if you play him right, he's the strongest character in 3v3's. He has a counter to most of the 'op' champions in the game, and his spells are perfect for the twisted treeline.
First off, his bandage is very handy on this map. Almost all the brushes are straight or close to straight, so a bandage passes through them. There are also tons of walls he can hop over using bandage toss, whereas on summoners rift there are only a handful. He has a perfect gank escape as he can fake one way, bandage to any minion or creep set and escape that way, or he can stun the closest ganker and make a run for it.
His tears are also one of the strongest skills in the game. With a slightly offtank build, you can easily hit 4-5% a second, and since it's area is so large, it works while chasing or being chased closely.
Tantrum is easily the best, as you can tank very effectively. It reduces the cd every time you're hit, so it becomes very quick when being focused, and does massive amounts of damage in an aoe. It also reduces 10 damage from basic attacks, giving you a slight edge. This means that mid game, all except super minions do 0 damage to you, and refresh your tantrum very fast. This lets you clear and farm insanely fast. The easiest way to amumu is be clicking e 98% of the time. You can jungle while texting, just mash e.
Ultimate: Is insane. Easily the best 'snare' as it's effectively a stun. 2 seconds over a large aoe with a very nice ap scale, it does a lot of damage and leaves the enemy team smacking thier keyboard. If popped when your team is low, it becomes the most clutch ulti
Countering famous enemies: Lee sin is probably amumu's worst nightmare, but even he's dealt with fairly well. Just bandage him when he shields, and in a team fight, he gets ruined in seconds. Darius is one of the most whined about characters. All you have to do is wait till you dip below 1200 hp late game, bandage him, throw a tantrum or two, and if he's still not dead, ulti and he will be. Teemo is also hated by most people. Teemo is absolute garbage against Amumu. If you know where he is, nonchalantly walk up to him, with tears on, and tantrum. He'll run like Usain Bolt, and then you bandage him. His sprint will be down by that point, so you can keep his speed and walk/tantrum his way to the grave.
Proof? I'm not that great of a player, I have 1300 ELO on 5v5, and as soon as I started 3v3'ing, I went on a 5 win streak with him. So far, I have probably 30 3v3's with him and have lost 2. This is my match history, and you'd need to look 5 more matches down to find a loss on the TT
That's my saved history. Amumu is just too strong for TT.