First you need to set up 3D hardware acceleration for your graphics card. Intel graphics users have it easy, you need not do anything. Although UT under windows supports software rendering, I could not find the option for that here.
There are several installers available under linux for UT, and I have no idea which one will work best for you. Try them all is my only advise. I used ut-install-436-goty.run available here.
Some other installers are available here and here.
Make the downloaded file executable, and launch it as root user (sudo ./filename with ubuntu).
The first two files I linked to allow installation of only the game engine, which is what I needed. Uncheck the "data files" option for that.
Now, we need to link the windows data files, I used this guide for that. Use these commands -
for i in Cache Logs Music Textures Help Maps Sounds Web;\
do ln -s /path/to/windows/ut"$i" /usr/local/games/ut/"$i";\
for i in $(ls *.u *.int *.ini *.dll);\
do ln -s /path/to/windows/ut/System/"$i" /usr/local/games/ut/System/"$i";\
Replacing the path with the one appropriate for your system.