I believe your cpu has enough power to handle dx11 in bf4, which is more optimized than mantle anyway, so i wouldn't worry too much if i were you.
It's probably a driver software bug. Try 14.3 beta (from 14.1 beta and onwards, all drivers should support mantle properly) or wait for the next driver release.
You do have installed the latest game patch, don't you ?
gigabyte X58A-UD3R rev.2, core i7 975 3.9ghz (stock voltage), windows 8.1 64-bit, corsair 12gb DDR3 1600mhz 7-7-7-20 RAM, xfx HD 5970 + gigabyte HD 5970 (stock clocks), coolermaster ultimate 900w PSU.
amd catalyst 14.7 RC3 aug12