I live in Durham and work at Duke. The area around here can be flat, but you can also find a lot of short steep hills, a surprising amount in fact. It's more hilly in the North/West of Durham than the South/East. I can't say that I've never had problems with rednecks in their pickup trucks, but I can say that I've had less problems than in tender Midwestern suburbia. The cycling season here is all year, though winter can get into the 20's or lower (F) and summer definitely get's over 100'F, even more so on the pavement. In addition to the road, there is a good selection of mountain biking, and Durham has it's own Fixed-Gear community as well.
I live in the center of the city, basically downtown, and I can reach good, sparsely populated roads in less than 5 miles both North and South. We have a regular bikers breakfast for commuters and you'll see a surprising number of people getting around by bike in their daily lives, at least for a North Carolinan city. The local shop has weekly rides and there are a few good cycling clubs to join. The Duke cycling club encourages non-Dukies to join us for rides and fun events like our own Fake Cyclocross Nationals with mandatory beer and donut laps.
Mountiain spots are further away, but not far; and just like mountains for road riding the Western part of the state is close and Tsali and Psigah are joys to behold.
Drop me a PM if you have any questions about where to live, work, eat (especially eat), and what to expect. I'm not a native Southerner but I really like Durham.