I've read several posts in which players seem to know what civics the AI civs are running, but I've never read how that is done. What am I missing?
For example, I have an early neighbor that I would love to flood with missionaries and get to convert to "my" religion, but before I waste the resources building missionaries, I want to make sure she's not in Theocracy.

Open a diplomacy window. At the top of that window there are icons (with mouseover flashcards) that show the civic set that leader is using.

B

If you go to the AI diplomacy screen with one of them, you can see little icons up above their head. Those are their current Civics. I think you can wave your mouse up there and it'll tell you which is which, if you don't have the icons memorized and don't have your book handy.

Wodan

Thanks, that helps a lot! :p