So, you decided hey a mummy that's kind of cute let's look at it's stats. WOW! What a health bar!
Too bad that's not why Amumu is truly amazing. I am here to tell you my play style it is similar to that of the Misunderstood Mummy located on this site, but I felt it needed to be updated and even brought more in depth. Now, PLEASE keep in mind I am not a good player by any means (which you will see later) I just really like Amumu.

Pros:
Tons of health
Good CC (especially the ultimate)
The tank you want on your team (eh maybe tied with Rammus)
Great DOT auras with sunfire cape and despair. If they attack you you also have thornmail basically they will be killing themselves even by getting within range of you.
Decent early game, SHINES late.

Cons:
Don't expect a ton of kills you are a tank. Except a LOT of assists.
Slow mover :/
Bandage toss not a very good escape method.
Essentially done for if ganked by 2v1 if not played well. Previously I just had this without the if not played well. Except I was ganked 2v1 consistently throughout the day. Have your ulti ready and you will actually probably get them both.
Slightly item depenedent when it comes to damage output.

So, expecting a lot of questions involving my rune choices.
Why not pick the runes that scale as you level in game?
Well my friend here's what I have to say to that.
Early game you love to see that first blood don't you?
Well, if you have +2.7 health versus plus 32 health early game I think you can see the difference.
Late game is derived from what early/mid game is made out of.
Ever hear the people raging about FED FED FED FED FED Yi is too fed we're f*cked!
Well, that can be avoided by the flat runes not so much by the scale by level.
Survive early game and you will get the kills or assists on the squishies (assuming you have a good lane partner)

Another question I expect: Why not just stack armor on all of them?
Now, I have actually considered this.
But, early game Amumu drinks his mana through a straw the size of Canada.
And like I said previously runes, to me, are early game differences.
So, I chose health regen and mana regen to compensate with the philosopher's stone early game you should be able to survive no problem.

Okay next up are the masteries.
I go 0/21/9.
Now in my 21 defense I CHOSE to skip the dodge chances. Why?
Because Amumu's tantrum when he is hit is dropped by .5 seconds on cooldown!
HOLY SH*T!
You can farm minions like no other. You can spam it in team fights for max damage output because well they will target you since you are the tank and you run in first. I mean it is what you are there for why not capitalize?
Another question I expect is why pick up willpower, well obviously to make it to where I can use cleanse more frequently. Personally I don't think the small percentage from ardor is enough to put more than I have to in it either.
Okay and I chose 9 in utility because the mana and health regen are nice and add to the runes and early game item build.
ALSO the neutral monster buff is EXTREMELY important. Mainly because you WILL grab golem and as a result you will be able to run with despair on almost all the time which is your main DOT dealer.

Okay guys these item choices are VERY situational. So, please pay close attention to this.
Okay so items depend on the opponent team. If they are caster/ability dependent champs you will stack up the MR with Force of Nature, abyssal scepter, and banshee's veil. If they are more physical dependent such as ashe, panth, and yi stack armor like thornmail.
I forgot to put in a health pot so be sure to grab that too with the regrowth pendant.
I can't begin to fathom how much I hate being laned with a guy/girl that forgot their health pot and are like SH*T I need to b.
I'm like "You're only level 2 what the hell man..."
Oh well solo lane exp I guess.
But anyways.
I get both philosopher's stone and heart of gold.
1) Good health and mana regen on stone to help you stay in the lane that much longer. Then the gold gain makes a HUGE difference coupled with heart of gold.
2) Heart of gold gives armor (obviously needed for tanks) and then it also adds to that gold bank of yours.

Lvl 1-6
So early game start with regrowth pendant and health pot. Next philosopher's stone (as SOON as you can get this B and get it!) So, you should be mowing through some minions with tantrum and possibly even grabbed a kill/assist on a squishy champ with bandage toss/tantrum/despair combo. You may be able to grab Heart of gold early game as well, if you're lucky.

Lvl 7-13
Mid game. So, now everybody has their ultimate and they will begin using it to their heart's like. Pick up ninja tabi. Why? ARMOR! Now I understand situations make it obvious that you should pick Mercury's Treads if they are heavy casters/CC. I DESPISE stuns, holds, immobilize, suppression, slows, etc when they are used on me because I can't do anything except absorb a little damage :/ So that's what Mercury's Treads and Cleanse are for.
Now that you have those Nike's to go with your stone and heart of gold time to start working on those other items.
Depending on the other team you can change the order I recieve my Sunfire Cape.
Next grab a Banshee's veil to counter those mages. Get the magic resist.
Get those two items and since they cost so much you will probably only get these and then HELLO late game!

Lvls 14-18
Okay this is where everybody team fights and wants to kick the **** out of the other team. Let's do this. Get Banshee's veil keep up that magic resist the health and mana are nice as well. Next get ahold of a Force of Nature. Now you have good armor and MR. So, the last item I get just for kicks and giggles Warmog's Armor. You've stacked the armor now add some Health to the mix and well you're invulnerable. Not sure if I want to move this somewhere else like before the hourglass. You're call I'm still deliberating myself. I also, put a few other options on item depending on your PERSONAL play style. Take your pick.

So situational items, thornmail over Force of Nature. If they are heavy melee/regular attack damage dps.
As expressed earlier Mercury's Treads over Ninja Tabi if heavy CC.
Oracle's elixir (located in consumables) allows you to see hidden units. So, if they have an Eve, twitch, teemo, or Akali on their team you best be buying this. You COULD use vision wards for your whole team, but eh I have to actually carry those...

Okay so after careful deliberance and a match or two of playing I have decided to edit this one.
Previously I had maxed bandage toss. Bad idea :/
I do however still pick it up for the stun to help, but don't level till far later.

First I like to maximize tantrum for damage and group killing. Helps with the push

Then despair to maximize total damage output with sunfire cape.

Last bandage toss. As this was brought up as low dps and only there for stuns and initiation.

Curse of the sad Mummy-Group CC. Rape follows. Be sure you're team is with you.

This is the hardest part to me. I personally am in love with the flash/cleanse combo.
You get heavy cc'd rather quickly as a tank. So, pop that cleanse flash out congratulations you survived that gank.
Now, I know a lot of people that don't like this and either go flash/ignite (I hate ignite personally) to get the extra damage.
OR
They go flash/exhaust. Now exhaust with amumu actually isn't that bad of an idea (I just can't give up my cleanse). You bandage toss up and go through the rotation like usual and then exhaust when the stun is gone continue chasing exhaust wears off they are at probably 1/4 health by this time guess what your bandage toss is recharged grab and finish.

PLEASE keep in mind that I said I am NOT BY ANY MEANS good at this game. But, feel in the hands of a good player (or at least better than me) this build has some pretty great potential.
Let me know what you think! No raging in or out of game! And please have fun :]