I have two PCs at home, one with an AMD card and one with an NVidia card. I'd characterize the NVidia card on Linux as "Windows-level hassle" (i.e. small, but present) while I think about the AMD card about as often as I'd worry about USB mouse compatibility (i.e. haven't given it a thought).
EDIT: small edge to AMD for graphics cards, but both are fine if using a distribution like Ubuntu
Yeah, like I said with my own anecdote I also didn't experience any issues w/ AMD graphics cards. Actually after searching a bit maybe I had that backwards and it was actually Nvidia that was the problematic one. I haven't had problems with either, but have read about problems and I thought it was AMD, but perhaps I was mistaken.
EDIT: small edge to AMD for graphics cards, but both are fine if using a distribution like Ubuntu