A possible build for Jungle Amumu; CC welcome.
For summoner spells:
I have not tested this build without rally, but it's theoretically possible. Smite/Rally is the most efficient jungling build but it's somewhat selfish. Amumu is a great carrier for many summoner spells: Exhaust, Rally, Fortify, and Clairvoyance, to name a few. Cleanse, Flash, and Ignite are all viable on him but generally not necessary. He's a beastly tank with this build, so long as you're not off pushing towers by yourself (Hint: that's one thing you can't do. Your autoattack does like 2 dps) you should be fine.
Boots: Consider the enemy team. Ninja Tabi is better for jungling, Merc Treads are great for CC-heavy teams, and each protects against a different damage type.
Aegis: If there is another viable Aegis-carrier on your team, let them take it so you can rush your godly Rylai's. Otherwise, build Aegis first.
Once you have your Boots, Chalice, Aegis, and Rylais, the game is probably near over or at least in lategame; the rest of the build is a list of luxury items that have different benefits. Use your brain; look at the strengths and weaknesses on both teams, and itemize to compensate.