Two Chefs (downtown and Pelham) - winner of Fall For Greenville's Silver Spoon for something like 5 years in a row.
Trattoria Giorgio (make reservations, and dont expect to be able to substitute ingredients!)
Thaicoon in Greenville near Furman offers some good Thai food as well.
Also, if you are looking for a quick bite of the drive-in/barbecue variety, Cookout restaurant in Spartanburg is an awesome place to stop by. As the name would imply, they offer typical cook out fair of exceptional quality that tastes like something you might grill at home. It's also great for anyone on a budget, as $6 should get you more delicious food than you could ever hope to finish. I really dont know how they make any profit.