Hi processor,
The GroovyArcade distro is meant for ATI cards only.
Now, what we know from Windows is that from the X series, only the lower ones work fine (X300, X600,...). The others (X1300, etc.) can't do low dotclocks. HD 2xxx and HD 3xxx have the same problem. This doesn't mean that they don't work: you can just setup GroovyMAME to use higher dotclocks and it will duplicate xres in order to do that, but that's kind of problematic with other emulators.
As this seems to be driver related, and not a hardware issue, in Linux, they might work just fine indeed. The problem is that we don't have information about it. It would be highly appreciated that someone with a bunch of different ATI cards tried the GroovyArcade distro and did the right tests to determine which ones are fully compatible or not. I've tested it myself with R 9250 and HD 4350 and it worked great with those.