Will my CPU be a bottleneck for a 7970?

no, generally it will be enough as a 3 ghz phenom 2 will be able to supply the cpu for most games of course some games like skyrim just don't work well.

why, because skyrim is a CPU-based game?

skyrim is badly coded and is very cpu intensive to the point that you need a 4.5 ghz i5 to have it not go below 60 fps no matter what graphics card you have.

its very cpu intensive and it only used 2 threads.

ive never had an issue with skyrim

and go for sandy bridge-e, ivy bridge is basically a 2400,2500,or 2600 i series chip with a smaller die size  

i see. i'm playing skyrim on the 360 anyway, for recording reasons. this encourages my upgrade

since you have a 6 core, the recorder should use the avalable cores not used up by the game

and Skyrim is still 32 bit, it isnt that cpu intensive

No,because Skyrim is more sensitive to clock speed than the no of cores.Also it performs better on any Intel cpu than AMD.
So there will be some bottleneck in a few ill-optimized games but you have nothing to worry about because the frame rates will be much higher than 30fps despite of bottleneck if you have a good GPU.
Therefore I suggest that you upgrade to 7970 becuse gaming depends mainly on GPU not cpu.