A dual core processor at 3.7GHz shouldn't limit you too badly even with a graphics card as solid as the 570, but it will limit you to some extent.This article
is a little dated, but it has benchmarks using both an E8400 (OC'ed to 4.4GHz mind you, but should be a nice frame of reference) alongside an OC'ed i7-920. The 5870 and 5970 are on there- the 570 typically falls between the 2 in performance. Make sure to take a look at the resolutions you are playing at (I'm assuming 1920x1200) and you can see that at higher resolutions, the E8400 is really hanging within a few percentage points of the newer platform.
The 570 wasn't out then (in fact the 470 wasn't out then either), but dual-GPU cards are the only ones that are being limited by the CPU at higher resolutions.
Bottom line, if you're looking at straight value you will probably be disppointed by the return on investment from upgrading your platform if you're going with a single card. At the same time, since we're talking about value the 570 may be a bit of overkill- there are some fantastic deals out there on 470's (which are still great cards) as well as 460's, 5850's etc. I'd recommend doing a little homework- check out the types of games/resolutions you're playing and see if a less expensive (but still very fast) card will keep you happy.