I live in Durham, but ride down into Chatham County a few times a week on the American Tobacco Trail which runs from downtown Durham to a point 4-5 miles east of Pittsboro. The ATT is part of the East Coast Greenway and connects with the impressive trail network throughout Wake County. Fearrington Village is a nice place, and there may be some nice rural roads that are nice riding. Unfortunately, everything seems to require interacting with US 501 (which is no longer the sleepy 2 lane road that it was when I first moved here.) I'm sure you'll find some reasonably good routes once you get to exploring.
I ride MUPs mostly, as well as urban streets. I've gotten away from road riding, but I do know that there are a lot of country roads that are quite nice and do see a lot of cycling activity, especially around Jordan Lake.
There is a heck of a lot of residential housing going up in eastern Chatham County - sprawl from Wake County! This may mean better infrastructure for bikes ... eventually ... but for now, it seems to mean more traffic on the country roads and more shopping centers.