The Foothills Parkway south of Knoxville is very scenic, affording plenty of mountain-top views towards central Tennesse and to the east into North Carolina. As a bonus, it ranks fairly high as a fun drive especially when one takes into consideration how much less trafficked it is than to the nearby Great Smoky Mountains National Park. The main problem with this drive is that it's fairly remote and I can't in good conscience recommend someone drive several hours to get there just for that driving experience alone. It's nice but not salivatingly so. I'm sure there are better drives elsewhere in the state's back roads.
Unlike the Foothills, the stretch of I40 from Newport (east of Knoxville) to Asheville, NC map is much more accessible and even more fun to drive assuming you can escape excessive traffic. Also notable is I75 from the Kentucky border southbound towards Knoxville pic1. I say southbound because all the downrange scenery is on this side and the highway is so wide in places that you'd miss much of it going the other way. The road conditions were very good as of January '01 and traffic has always seemed manageable.