I'm running a 32MB Radeon 7200 PCI with the Omega drivers (v2.5.97a). I have an Athlon 1600+ on a WinXP SP2 system and 768MB of SDRAM.
I can run CivIV, and have played through a couple of games. However, I run it on the lowest graphics settings possible. Aside from many of the other bugs experienced by people with waaaay better cards than I, it works for the most part.
I will say, though, that when zoomed in close enough for unit movement and city view, the framerate slows down. A lot. I'm a pretty patient person, so I just wait for the transitions to happen. I occasionally get the Chesire cat leaderheads, though most of the time even those work.
Zoomed out to globe view and it's just fine.
I've only tried playing on a small and regular map, limiting my competition to 4 other civs (so far). Knowing that the game is still buggy and stressing my system (memory leak??), I haven't wanted to play on anything larger or more demanding until I've patched.
Waiting for the patch would be good. You might be taking a gamble if you just go out and buy it -- obviously I don't take my own advice

, but whatever.
After a patch I can try to let you know if performance improves for me, if you'd like...