I decided to build a PC with a 5970 a few days ago, and came up with this:
Intel Core i5 750
Gigabyte GA-P55M-UD2, Intel P55, LGA1156, mATX, CFX
Antec Mini P180
Sapphire HD 5970 OC
Thermaltake Toughpower 750
Mushkin Blackline 2x2GB
now my brother said it would be able to run just fine...