http://bugs.winehq.org/show_bug.cgi?id=3498
Specifically, post 24
$ sudo su -
cd ~user/.wine/drive_c/"Program Files"/"Firaxis Games"/"Civilization III"
chown root . civ3PTW Conquests
chmod 1775 . civ3PTW Conquests
ln -s LSANS.TTF LSANS.FOT
ln -s LSANS.TTF civ3PTW/LSANS.FOT
ln -s LSANS.TTF Conquests/LSANS.FOT
I was able to get the Steam version of the game up and running (changing directories appropriately to find the root of Civ3). That said, I've never played any version of Civ on the PC before, so I can't vouch for how well it's running, but I did create a Warrior before quitting to do other stuff.