- iMac (Mid 2007 or newer)
- MacBook (Late 2008 Aluminum, or Early 2009 or newer)
- MacBook Pro (Mid/Late 2007 or newer)
- MacBook Air (Late 2008 or newer)
- Mac Mini (Early 2009 or newer)
- Mac Pro (Early 2008 or newer)
- Xserver (Early 2009)
That covers all the recent systems, but for ones that are slightly older, it's being reported (unofficially) that an updated graphics architecture intended to smooth out performance in OS X's graphics subsystem is the underlying issue. It's no coincidence, then, that the unsupported GPUs happen to be ones that were fairly common back before 64-bit support became mainstream.
Mountain Lion apparently doesn't play nice with 32-bit GPU drivers, and while Apple could spend time and resources bringing older models up to par, the Cupertino company decided it was better off dropping support altogether. That's a tough pill to swallow for owners of Mac systems that are barely more than three years old.