AMD has made serious commitments to their open-source drivers. They've released documentation for their chips and have several people working full-time on the OS drivers.
As for supported chips, everything up to "Northern Islands", the last generation, works really well. The latest "Southern Islands" chips also work well but don't have all the features implemented yet.
I use a HD7970 in my workstation. 2D works great and is very fast. 3D is pretty slow, b/c many things aren't accelerated yet. So, for example, minecraft runs about 20fps.
The next generation of AMD gpus "Sea Islands" will use the same drivers as "Southern islands" so the former will be supported as soon as the latter.
Probably the biggest issue is power management. dynpm dynamically changes the clock but doesn't work with multiple heads. The "profile" power management flickers when changing the clock speed.
The reasons for the lack of support for "Southern islands" is that support required many changes that couldn't be release incrementally. Better support is expected when the "Sea islands" chips come out. Check out this post by an AMD developer for a little more detail: http://phoronix.com/forums/showthread.php?73824-Bridgman-Is-...
Check out http://www.x.org/wiki/RadeonFeature for a table of chips and features and information about mapping engineering names to board marketing names.
If you have never experienced kernel mode setting(KMS) and in-kernel drivers I would _highly_ recommend it. It is such a better experience. It's so smooth, switching from xorg to a VT doesn't flicker. And having to upgrade proprietary drivers is a huge pain. I could never go back to UMS and proprietary drivers.
As for my recommendations, if the intel HD3000 or HD4000 meet your 3D needs, they work extremely well and have great in-kernel drivers.
If you need faster OS 3D or a discrete card, go for a "northern islands" Radeon. bridgman recommends a mid-range card for the biggest band-for-the-buck with OS drivers. The high-end cards won't be slower but the drivers aren't as optimized to get the highest performance.
If you are patient and like submitting bug reports you should get a HD7970 ;)
EDIT:
Oh and if you need OpenCL on gpu, intel doesn't support it yet. I believe its works on AMD "northern islands" and earlier.
For more info check out:
http://software.intel.com/en-us/articles/opencl-sdk-frequent...
http://www.phoronix.com/scan.php?page=news_item&px=MTI3M...
http://www.phoronix.com/scan.php?page=news_item&px=MTI1N...