Depending on how much time you have to spend seeing sites and doing activities while you're in transit, I recommend trying I-66 west from DC and taking the I-81 Blue Ridge route through western Virginia to Knoxville and on to Nashville and skip Asheville. Unless you are planning it as a gateway to spending time in Great Smoky Mountans NP (or you must see the Biltmore Estate), Asheville isn't worth going any direct route you'd take to get there from DC, and then from there to Knoxville.
The Blue Ridge route has loads of opportunities for activities, site seeing and natural beauty along the way that you won't get close to touching the other route.
You can still go to Asheville if you go this way, but it's a back and forth of about an hour and a half each way on I-40 to go and then get you back on track.
"If all mankind minus one, were of one opinion, and only one person were of the contrary opinion, mankind would be no more justified in silencing that one person, than he, if he had the power, would be justified in silencing mankind." - J.S. Mill