The first thing I'd recommend is more RAM. Civ III uses a LOT of RAM just to start (some people have seen total memory in use by Civ more than 128Mb, IIRC). Considering how cheap 128Mb of RAM is right now (i.e., the cheapest it's *ever* been), this would not only help you with Civ but should pay dividends with a lot of other apps and games.
If more RAM is out of the question, at least defrag your disk, since the OS is constantly thrashing out to virtual memory to play Civ. And for that matter, close everything else (including tray apps) that might be claiming RAM. WinAmp, for example, leaves a 900Kb app running even if you think you've closed it (I think it's called the WinAmp Agent, or something). For someone with 256 Mb of RAM, that's not a big deal, but for someone with 128, it can make a difference.
Secondly, make sure all your drivers are updated and you're using the latest Civ (v1.16f) patch. Latest DirectX, etc. These things are no-brainers but a bad driver can slow the game down immensely.
Third, if you're dragging that heavily between turns, use the "don't show computer moves" switch (v1.16f only), it does make a difference, especially in later gameplay when there are a lot of enemy units moving around.
Fially, it's an unavoidable fact that the more civs you play with and the larger the map, the slower the wait between turns will be. On a 1.5Ghz P-IV it shouldn't be so bad, but I played the game quite extensively on a PIII-500 at work and it can get very slow with 16 civs on larger maps. This is the unfortunate tradeoff of having a better AI.
Dan