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I-40 to US-23 is the easiest direct route. I-40 winds through the hills but there are only a couple of grades that are over a 1/2 mile long. Much of US-23 is four lane down to Franklin.
If you're driving or pulling an RV, your best bet is probably to take I40 east into NC, then south on US74 (Great Smokey Mountain Expressway) to Sylva and then take US 23/441 south to Franklin. Lots of up and down, of course, but nothing excessively steep and good highways.
I haven't driven I24 south from Nashville, but I do know that some sections of US 64 through the mountains are pretty steep and tight.
I travel to Franklin NC from Nashville TN at least once a year, and often twice. I've tried 'em all and prefer I-24 to I-75 E, then exit at exit 20 onto route 64. It will take you into Franklin. There is a pretty good grade about 40 miles after Murphy but I've never had a problem and I pull a 30' TT with a F-250. We live about an hour out of Nashville and it takes us just under 6 hours. Oh and you'll experience some narrow roads next to the Ocoe river...but take it easy and it'll be fine.
If you are have a diesel, I recommend stopped in Jasper TN, exit 18, I believe its a BP. Works for us!