I get the impression that many people here don't actually bother, but still get much better results than I do. I'm probably a borderline OCD sufferer, and on most turns I check every city to make sure the tiles are allocated optimally (eg. maybe it's only 1 food short of growing this turn, in which case I'll temporarily drop a little bit of production in order to get enough food to grow) and everything looks right. I pay close attention to the number of turns of slavery anger left, and just try to plan ahead in general with each city. I also look at what techs my rivals have (every turn; it's really important). I need to know who has the techs needed to build any wonders I want, or who's getting close to liberalism, or who is going to end up building the AP or UN, that kind of thing. Also, it's worth checking to see if any of the backwards civs are feeling rich - sell them tech before soemone else grabs the cash! Less frequently, I check the resources screen to see if anyone has anything useful, or can spare me any money for my surplus goods. Occasionally I'll reshuffle my trades to get better deals.
At the start of the game, I try to check the diplo screen every few turns to see if anyone is able to offer me open borders yet (I don't speak to every single leader, but I'll check to see if there are any open borders between other civs, so I know who has writing). Later, I keep checking to see if anyone has reached alphabet before me.
I check the power graph very regularly (it's crucial), and the other stats less regularly (GNP is worth checking). Very occasionally, I'll glance at the victory conditions screen - the domination section is the most interesting bit. When someone builds a wonder, I'll try to find out which city it's in by checking the top 5. It's also worth checking to see if anyone has missed the wonder and ended up with a heap of cash to buy my old techs. On BtS, I occasionally glance at the "info" part of the diplo screen to see what kind of civics people are using, and who will be giving each other fave civic bonuses.
To be honest, I look at a lot of the info screens compulsively even if they're not useful; I just like to have a feel for how the world is shaping up. I look at the top 5 cities screen much more often than anyone could ever need to, but it's nice to imagine the world's citizens speaking with awe about the Seven Wonders of Thebes, or the Three Shrines of Mecca.
Like I said, I imagine most people don't check things regularly each turn. In summary, I always check the tech screen and cities, often check the resources trade screen and power graph, and occasionally check everything else. I'd be very interested to hear what other people do...