So far, the biggest problem over at the CivFanatics Forums is with (stop me if you've heard this before) drivers for ATI cards.
As for myself, I got it to run okay on my desktop (with recommended specs in everything except my ancient 64MB GeForce 3) with a just little skipping in the intro video. (Thankfully, the in-game movies work fine.) After about 4 hours of playing, I'm impressed. The only real negative so far is that the Civilopedia has taken a major step back from Civ3. It's not searchable, and everything (techs, buildings, wonders, etc.) is indexed by the in-game symbol rather than the name.
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Look forward to reading them.
Much appreciated, actually not as bad as I though one extra memory stick and a DVD rom drive.