I experienced two majors problems with Civ3 on Vista:
*It occasionally crashed a couple turns after loading the game, for no apparent reason.
*Saved games sometimes didn't actually save.
Once one of these problems occured, they both occured and neither one would go away until I reinstalled Vista (reinstalling Civ3 did not solve the problems). Then it would run fine for a couple weeks until the problem started again, at which point I'd have to nuke and reinstall Vista again.
But my sister hasn't had any trouble playing Civ3 on Vista. It works for some people, for others it doesn't.
I've long since returned to XP (on the same machine), and Civ3 is happy as a lark. Not sure what the problem with Vista was (I tried everything to solve it), but if you're a Civ3 fanatic, I have to specifically recommend XP and not Vista/Win7 (unless you already have a Vista/7 machine that runs it fine). On the other hand, Civ3 is only $5 on Steam, and it seems most people either have no problems or minor ones, so even if you're running Vista I'd have to recommend buying Civ3 for $5.
As for support, I e-mailed Atari after I'd run out of options on the Web and CFC, and they did reply, saying that Civ3 is not supported on Vista. In most cases it runs well enough, but if you're unlucky like Salarakas or I, you're out of luck.