Mass Effect, through Steam, has only one problem on CXG Linux 8.1.4 (and on Mac too, apparently): the mouse. And here is how to fix it (this is entirely based on Exile's post, thanks, Exile!).
(You can also try downloading the precompiled binaries, at the end of the post.)
The original dinput.diff no longer works on wine 1.1.35 (the latest as of post date), so I've created a modified version.
sudo apt-get install build-essential patch
tar jxvf wine-{version}.tar.bz2 cd wine-{version}
patch -p1 < winex11.drv.diff patch -p1 < dinput.dll.diff
If you stuck to the original dinput.diff, now is the time to finish the patch by hand.
./configure (or ./configure --without-freetype, see paragraph below) cd dlls/dinput make depend make cp dinput.dll.so {MassEffectFolder}/Binaries/dinput.dll cd ../winex11.drv make depend make cp winex11.drv {MassEffectFolder}/Binaries/winex11.drv
A note on configure: I could not run it successfully in amd64 because there where no 64-bit libraries for freetype, or something like that. So I had to configure without freetype. There's probably all sorts of options that this compilation doesn't need, actually, but I haven't figured them out. The ./configure command will be the most time consuming, but should take only a couple of minutes.
Naturally, all commands should exit successfully.
Your {MassEffectFolder} should be in /home/{user}/.cxgames/{bottle}/drive_c/Program Files/Steam/steamapps/common/mass effect/ (pheww!).
Precompiled amd64 Binaries (ubuntu karmic, wine 1.1.35)
Precompiled i686 Binaries (ubuntu karmic, wine 1.1.35)