There are so many possibilities and variations that I hesitate to select a few to mention, but I think I have been on most of them so I will mention one or two.
You could take US 19/74 west from Asheville. Then stay on US 74/23 near Waynesville, NC. Follow all the way through Andrews and Murphy, NC. US 64/74 to intersection with I-75 to get through Chattanooga. Then I=24 to US 64 all the way across southern Tennessee to Memphis.
It is far less complicated than it sounds. Just take a look on a good map of North Carolina and Tennessee. This is not a fast route, but a pleasant one, and if you have time there are some beautiful spots along the North Carolina portion (Nantahala Gorge, for example; slow driving but beautiful)
There are possibilities west from Memphis, but my comments are complicated enough so I will stop right here.