I'm not sure.....it has been a long time since I ran UT on a nix box. It was UT2003 on RedHat 7.2 last time I tried it.
I don't think the Windows folder copy will work on the Linux kernel since the Windows (C:\Unreal Tournament\System) has ".exe" executables for win32, where as, the linux executables are normally ".sh" or no extension w/ a chmod of 755 (RedHat highlights executable files in green).
The game itself is written in a hybrid of C, which Linux is built around. They way the game is called in the linux version should be ".sh" and not ".exe" if that makes any sense?? That was the way it was done in UT2003. I wonder if unrealadmin.org has any info? I'll see if I can dig it up.
Here's some Patch Info for UT2004 on Linux
http://downloads.unrealadmin.org/UT2004/Patches/Linux/
This guy has a screen-shot how-to for the general installation of UT2004 on Linux
http://www.mepisguides.com/ut2k4/ut2k4.html