What games would you recommend, having played and immensely enjoyed Sid Meier’s Civilization I, II, and III. I don’t have a high enough spec PC to play Civ IV so older game suggestions would be appreciated.
Have you tried the older Age of Empires games? The original is a blast, and I usd to run it on a Windows 95 PC. AoE II should run on pretty basic spec. Not sure about AoE III but the first two should be fine.
Europa Universalis 2 is ace, and it’ll run fine on a low spec machine. Europa Universalis 3 has similar requirements to Civ 4 though.
Have you played Alpha Centauri/Alien Crossfire? AIUI it’s got a very similar engine to Civ II, and has a nice overarching story to go with it.
Galactic Civilizations is simply excellent when it comes to turn based strategy.
The AI will knock you down a peg or two as well. It doesn’t even ‘cheat’ until the highest two difficulty levels so when you’re getting your ass kicked on medium you know it’s your own fault for not planning properly and not just the AI getting extra resources.
This is one of the strangest simul-NewTopic pairs I’ve ever seen. I posted an almost-identical OP a few hours before this one here. I guess we’ll get twice the number of suggestions. 
“Ceasar III” a Rome-themed city building/resource management game. Build Aquaducts! Trade with the Phoenicians! Fight off barbarians! Appease Poseidon! It seems to often come bundled with “Pharoh” a similar game based in Egypt – but there are some notable differences between the two, its not the exact same game with different graphics.
You might consider the Total War series. While the Medieval part of it is the penultimate world control / empire building part, I certainly don’t scoff at Rome: Total War. (In fact, I bought it just before Christmas and enjoy it immensely) It should also work with a computer that can handle Civ III.
It sets you in control of one of the three primary Roman factions, to make your way to domination through whichever means you want. While its battles are a bit more involved than those of the Civ series, it’s still a numbers game and you can play through the entire thing without taking direct control of battles, if that’s what you wish. (It has to be admitted that they have focused on battles and tactics to the detriment of the strategic dynamic, though)
Master of Magic, if you can find it. It’s like Civ with a fantasy setting and there’s an optional RTS mode for battles. If you can find it, you might need DOS Box for it.
I second this one. I can’t recommend it highly enough. You really should get it.