You just wanting to "get through" or you want to stop and see a few things?
Leaving the west end of Cherohala Skyway at Tellico Plains, continue west and a bit south to Chattanooga, visit Lookout Mountain. Leave Chatt going south on I-59, you can clip the corner of Georgia to say that you visited the state and color it in on your "states visited" map.
Turn west, go across the tops of Alabama and Mississippi. Huntsville, AL has a nice space center tour/museum.
If you stay at the north border, you will end up in Memphis and visit Graceland, home of Elvis Presley.
Head northwest out of Memphis on US63, head toward Walnut Ridge, pick up US412. Not a "technical" road, but most of northern Arkansas is just pleasant riding through scenic areas. Once you get west of Springdale, you may as well slab it, there is nothing to see until you get to Colorado, unless you want to swing north into Kansas and visit Dodge City.
Leaving Denver, feel free to slab it west, that section of the freeway is downright scenic. As you get past Grand Junction and Rifle, the highway is in a canyon, so it is a double-deck arrangement. Turn north at Fruita, take 139 to Dinosaur, then clip the corner of Utah so you can color in that state, too. Continue past Flaming Gorge to I-80, then west a bit to US30, then go north on US89, US189 and US191 to Yellowstone. As you go north, you are litterally just a few feet away from Idaho (hint, hint). Go north out of Yellowstone, catch a bit of Montana then go east.
Mount Rushmore and Crazy Horse are on the way to the Black Hills.
After the Black Hills, there isn't much, but it would depend on your route preference. Stay north, end up in eastern Wisconsin, take the ferry across to Michigan. That will avoid Chicago, always a big plus.
Once you get east of Toledo area, northern Ohio is pretty flat until you get to Pennsylvania and New York. US6 across Pennsylvania has a bit of history. Lots of forests, rather scenic. New York will have the Finger Lakes area, but most of those roads are north/south, so you will have to skirt them on the north or south ends.
Lots of things there to make your three-week trip just a teaser, these are just a few ideas.