Most of these climbs are not going to be tri-bike friendly, as you have to get back down after you go up. Sinuous and full of turns would be an apt descriptor. Even the milder ones are still better tackled on a road bike, but perhaps that's just me.

Also, don't confuse lower traffic with safety. Rural roads generally permit 50 (if not 55 mph) on two-lane, shoulder-less country highways. You may not see a lot of cars, but if something goes awry, it won't go well.

Having said all of that, the region of Virginia on either side of I-81 would get my pick. The highway sits in a broad, rolling valley abutted by two ranges (Blue Mountains to the east, Massanutten to the west. All the climb you could ever want (try Wintergreen, if you really want steep gradients), and there's even the Skyline Drive / Blue Ridge Parkway (generally follows the ridge of the Blue Mountains), which is probably the most conducive for tri-bike climbing. The roads from the valley to the mountains (the ones that traverse mountain gaps) are less conducive for tri-bikes.

How much money you got? In SC by Greenville there is the Hotel Domestique bike themed place with bike tours. Plenty of good riding out that way and Greenville is a decent little city with good eats and brew.

I had to work down there a few times and rode out of town a bit. Not bad.

I liked getting mussels at the Trappe Door and washing it down with some good beers. If in season you could also catch a baseball game in downtown or something.

+1 on Greenville. I was there for Hincappie Gran Fondo and the riding was just awesome.

Another good option is Charlottesville, I've ridden west of it and is also great. You have the BRP and Skyline drive. Wintergreen and Vesuvius climbs.

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Northern Florida + Some Climbing = Null Set. :) But you can find a lot of lower traffic roads. Still, I think you could do better in NC/SC or Northern GA.

There are some good roads to ride on in southwest VA with plenty of climbs and not too many technical descents that are more than doable in a tri bike. you got the blue ridge parkway (and if you go a bit more north skyline drive)

I used to ride in this are all the time on my tri bike and cars were much more friendly than where I am now