Coupe of suggestions;
Strategically - Mountains are either much up-and-down or else lengthy convolutions along meandering rivers. If time/effort is an issue, you might want to go west until it's flat, then southwest.
The Adventure Cycling Folks have a great rep. You might want to check out some of their maps. Not just for this leg, but for the whole trip. They mapped out the TransAmerica Route and have otehr coast to coast routes.
The ACA "Underground railroad" route might be close to where you want to go to get to TN. This also follows the "West until it's flat" strategy.
If you go "skyline" you could ride the skyline drive/blueridge parkway until you get tired of the hills, then drop into the "Great Appalachian Valley." Then maybe switch over to the cumberland valley at some point.
http://en.wikipedia.org/wiki/Great_Appalachian_Valley
Colonials used muscle power to get over those hills, too.
http://en.wikipedia.org/wiki/Great_Wagon_Road
My guess is that the Transamerica route is farther north and hillier than you want.
Just some thoughts.
Write and let us know how it goes.