This could be a FAQ.
I think there’s a route from Nashville (Brentwood?) that goes straight into Robbinsville, NC, but I’ll have to dig through my files on a different computer to see if I can find it. Until then, here’s a potential route through Chattanooga. I’ve ridden and/or driven perhaps 80-90% of these roads, and while they’re not always comfortable, I think most of the route is reasonably safe. Note I would NOT! recommend taking 64/74 from Cleveland to Murphy. Somebody did a trip report going that way (in Adventure Cycling, IIRC), but I regard cycling a dozen miles through the Ocoee gorge as pretty close to suicidal.
Anyhow, try starting with the Leiper’s Fork permanent. Sorry, while Nashville is supposed to have a good bike route, I don't know a way across town, so my suggestion starts south of there. Get off the Natchez Trace at Leiper’s Fork, coast down the hill to the east a quarter mile, and pick up the route:
https://ridewithgps.com/trips/4264197Somewhere around Christiana, pick up the Sewanee 300k:
https://ridewithgps.com/routes/9491110 From Sewanee to Chattanooga, you’ve got a few options. First, take 41A a few miles out of town, then drop down to South Pittsburg on TN 156, cross the Tennessee River and go up to Haletown, then follow US 41 into Chattanooga. There’s some busy traffic for 2-3 miles on the south side of the river coming out of South Pittsburg. The last 4-5 miles going over the foot of Lookout Mtn. has stead traffic (especially at rush hour or when there’s a wreck on the interstate), but they’ll usually give a cyclist the right-most of four (narrow) lanes. Second, take 41A to Monteagle, then pick up 41 and follow it into Chattanooga. There’s a fair bit of traffic through Jasper, but it’s mostly got a good shoulder and is fairly straight. Third, go up through Monteagle and Tracy City to Gruetli-Lager, then drop down to Whitwell, go over to Powell’s Crossroads, then over Suck Creek Mountain on TN 27 to US 127. This is perhaps the quietest road route into Chattanooga. If you’re interested, I can give you some cues to get from Dayton Blvd. into downtown.
From Chattanooga, you could take the proposed USBR 21 up to Madisonville or Vonore:
https://ridewithgps.com/routes/11494161From Madisonville, take 68 to Tellico Plains, then take the Cherohala Skyway and follow NC 143 to Robbinsville. If you want to go further into NC, I’d suggest going over the mountain to Almond (along 143?). I haven’t been that way, but the NC 28 route from Fontana Dam to Almond is a typical mountain road: narrow, winding, limited sight distance, and too much traffic for me to be comfortable. From there you’ve got a couple miles on the racetrack (19/74) shoulder to Bryson City, and 19 from there to Cherokee is narrow, winding, lightly trafficked, and mostly slow traffic. In Cherokee you can pick up the Blue Ridge Parkway.
Alternatively, go northeast from Madisonville to Vonore, cross the Little Tennessee River, and take TN 72 to US 129, then enjoy the Tail of the Dragon into Robbinsville. Here, you could take NC 28 into Almond, but see my warning above. No camping on 129, while there are a couple of campgrounds along the Cherohala.