I've been reading some of the benchmark articles but have no idea which tests matter more than others.

Here's my use case:

I run Ubuntu (whatever the latest LTS is) and would ideally run the open source drivers (hence preference for AMD). This is a media box which usually just runs XBMC and plays video through my 1080p TV. So the main requirement is hardware video acceleration. I also want to game on it from time to time, especially since lots of stuff is now coming out with Linux support on Steam these days. I don't need ridiculous performance, but it should be able to run recent games at 1080p at a decent framerate.

The other requirement is that ideally it is passively cooled, or at least very, very quiet. The case is pretty roomy (though a monsterously long card would not fit) with plenty of air flow from big chassis fans.

The existing hardware is fairly modest -- a Core 2 Duo whose speed I don't even remember, and the slot on the motherboard is PCI Express 2.0 x16. (I believe PCIE 3 is backwards compatible, though I could be persuaded to upgrade the motherboard if it'd make a huge difference.)

Any suggestions would be greatly appreciated.