$350 wouldn't really be enough to upgrade to Core 2 Duo ($170ish for processor, $100-130ish for motherboard, $150-200ish for memory, plus more if your graphics card is AGP...), so I guess Pentium D and more memory is the way to go for now. However, I HIGHLY SUGGEST waiting if you think you can spare a bit more cash in the future ($500 will get you a nice Core 2 Duo setup).
Depending on what resolution you're running, the graphics card might be your biggest issue. If you're doing 1024x768, you'll be fine. 1280x1024 might get a little iffy, and 1600x1200 is pretty harsh on Oblivion for a 6600GT.
So, basically, is that PCI-E or AGP?