What you need to run "smoothly" depends hugely on the type of game (in particular, map size) you're playing, as well as what you consider "smooth". For AI turn times, the biggest factor (assuming you have sufficient RAM - at least 5 GB and a 64-bit OS) is single-threaded CPU power.
The 64-bit is key, as otherwise you'll be limited to 3 GB of RAM (2 GB without the 3 GB switch), which will be limiting on the largest maps and mods. Ideally, you'll have at least 4 GB for Civ4 + 1 GB for everything else, hence the 5 GB I mentioned above. It doesn't really matter if the 64-bit OS is XP x64, Vista x64, 7 x64, or 8 x64 - the key is that it's 64-bit. Even Vista will work if you don't leave tons of other programs running while you play Civ, and I say that as someone who hates Vista even more than Mac OS 9.
The single-threaded part is also key. Civ4 will not take advantage of multiple threads for AI turns (nor will any other Civ game, except perhaps Beyond Earth). If your choices are a 3.5 GHz dual-core or a 3.0 GHz quad-core, all else equal, the dual-core will be better for Civ4. Anything from Intel's Sandy Bridge generation or later (such as the i5-2500) is more or less equal here, as Intel's CPU performance, per-core, has not increased significantly in the past 3-4 years, and AMD has not yet caught up to Intel's huge lead. The current best CPU value for Civ, assuming you value time spent waiting on the AI, is probably the 4670K. Pair that up with 6 GB RAM and a 64-bit OS, overclock it a bit, and you'll be more or less as good as you can get.
Graphics cards don't really matter at all for AI turn times. Perhaps with some mods it might affect how fluid the game is during your turn, but generally anything that's new will have an adequate GPU, including integrated GPUs.
I'm skeptical that an SSD would make any difference, as provided you have enough memory Civ shouldn't have to load much from disk once the game's going. But I haven't tested that myself.
And yeah, whoever designed the minimum/recommended specs was a masochist. I started with a PC with 512 MB of RAM, and it was agony. Eventually it got bumped to 1 GB, but I still eventually went back to Civ3 since it was so much less agonizing to play on that hardware.