According to Everest, my AMD/ATI 5870 has 1024MB of memory. Playing at 1920x1200 resolution, with AA off, 1024x512 map texture size, no shadows. Using an i7 920 6GB RAM, Win 7 x64, latest drivers...
Not sure if this is related, but I also noticed my PC slowing down with a lot of hard drive access after a couple of hours (using virtual memory?), but this was resolved by exiting and re-entering the game.
None of this is a major problem right now; I just restart the game and carry on for a couple of hours. However, it might be a concern in future if more features are added and frequency of the problems increase.