I also found that Wine did not do a good job of guessing my graphics RAM, so I had to run wine regedit.exe and add the key HKCU>Software>Wine>Direct3D>VideoMemorySize and set the value to 512. Find the library called gameux, and add it to the list of overrides, and then edit it to be disabled. I ran winetricks msxml3 msxml4 vcrun2003 quartz devenum corefonts lucida tahoma, because I found it somewhere on the internet. You run Winetricks with arguments of various dlls and other things you need to be manually overridden. Next, you'll need a shell script called Winetricks. Export will save that variable in your terminal window until you close it, but with multiple terminals open I found it safer to just compulsively enter it. I then ran export WINEARCH=win32 before I ran any wine command. Since I wasn't using Wine for anything important, I just deleted my ~/.wine directory.
It is also easier to use Wine as 32 bit, instead of trying to get 64 bit to work. I have the actual discs, but found it much easier to download them. Also each disc is one independent installation. Wine doesn't work well with multidisc installations, but if you have isos you can mount them all before you begin. To begin, get iso's of the game and two expansions. It took me several hours, so I figured I'd document the process here for future reference. I've used Wine a bit, but not much with modern games like Civ 4. I have a week off, and figured I'd be productive by playing Civ continuously the entire time.