Quake III runs perfect on linux, but you don't want to run the Win32, you want to run the one for linux.
if you have Quake III installed in windows, just mount the partition, mine is /d/games/quake3
Then download this file: ftp://ftp.idsoftware.com/idstuff/qua...t-1.32.x86.run
Through a shell windows goto the directory you installed it to, and type: 'sh linuxq3apoint-1.32.x86.run'
For the install path type the windows quake3 install path (ex. /d/games/quake3 in my case) and say yes to making a sybolic link.
then you can type 'quake3' in an X console window, or make a link to 'quake3', and you are good to go.