I'm fairly sure the minimum for full HD playback is a HD4200. When I built my HTPC 1.5 years ago, I scoured the internet to figure out exactly what the minimum requirements were for HD playback. I ended up with a Athlon II X2 Regor 240 (2.8GHz) and a HD4200 since I was confident that would play back every HD format currently available. I'm fairly sure my CPU is slightly more than needed, but I think the 4200 is near the minimum needed.
Just found this to confirm that you need a HD4200 or better for full decoding support. - http://en.wikipedia.org/wiki/AMD_700_chipset_series#Integrated_graphics
If I were you I'd find an old PC with powerful enough integrated graphics. Adding a GPU will make the thing a power hog for what you want to do.