So in a couple of towns I've got some dead bodies lying about from various fights, plus some dead trolls outside of Bravil, and they don't seem to be going away. Do the dead bodies ever "clean up" and go away? Or will I have to drag them all to in inconspicuous place (thanks, canol in Bravil!)?

It depends which dead bodies you are talking about. Some of them never go away in the vanilla (non-modded) game; I know Glathir is one whose corpse always remains in Skingrad. Most dead bodies eventually do get 'respawned' out of the game, though.

If your game is not modded, the reason why bodies aren't disappearing could be because you're walking your characters by these bodies again and again. If you stay out of the 'cells' where the bodies are located, they will eventually vanish. This takes 3 days, though.

Okay, I'll give it a few days then! Glad to know I won't have to drag all the bodies out of the way.

Renee is exactly right. If interested, here is a list of all the 'permanent corpses' in Oblivion: http://www.uesp.net/wiki/Category:Oblivion-Permanent_Corpses