I can offer this:
$ lspci | grep VGA
01:00.0 VGA compatible controller: Advanced Micro Devices [AMD] nee ATI Radeon R350 [Radeon 9800 Pro]
$ glxinfo | grep direct
direct rendering: Yes
$ lsmod | grep ati
cpufreq_conservative 13598 0
$ lsmod | grep radeon
radeon 641550 2
ttm 46672 1 radeon
drm_kms_helper 26501 1 radeon
drm 128416 4 drm_kms_helper,ttm,radeon
i2c_algo_bit 12706 1 radeon
i2c_core 18989 5 i2c_i801,i2c_algo_bit,drm,drm_kms_helper,radeon
power_supply 13283 1 radeon
$ compiz --replace
compiz (core) - Fatal: Software rendering detected.
compiz (core) - Error: Failed to manage screen: 0
compiz (core) - Fatal: No manageable screens found on display :0.0
Launching fallback window manager
I got no idea about radeon at all (most of my machines are intel, one is with nvidia, but it's so old that i am happy that it logs into the GUI at all). I searched the web, but did not find much info what to do.
The German Ubunut wiki says it should support 3D.
(accept as is?, use the non-free whatever-ya-call-it ? creating an xorg.conf, if yes: someone has got a radeon one to offer? )