Some part of every cross-country Interstate needs repair, some part is currently being repaired or completely rebuilt, some part has been recently rebuilt and is real nice. Some places are a pain no matter what the conditions. There is a
federal information site that is kept up to date with respect to construction projects and emergency closures on the National Highway System, but it doesn't tell you where the road might be rougher than you'd like.
The Federal site links to state information sites, since the states are responsible for projects. It will not usually have really short term closures, like cleaning up after a wreck, closing a road as a crime scene, or when a tornadic storm closes a section of highway for a few hours.
I know Amarillo to Nashville. Most of the areas I knew about being repaired have been finished since I was last through a couple years ago. Eastern Arkansas had a 20 mile rough spot that was rebuilt from a new base (the road was being undermined in a flood plain); that took more than two years of detouring, but now it is done.
From western I-55 junction and on into Memphis is still a mess with construction. The junctions where I-40 and I-55 run together are a mess because of traffic volume even when there is no construction. It is a merge of two Interstates that are already bumper to bumper.
Coming from southern California I would consider I-10 to I-20 to I-30 working a shallow diagonal route to Little Rock, then I-40 to Memphis. That is a lower altitude, lower pass route across the Rockies. Another option is to stay on I-20 into Mississippi and take the Natchez Trace Parkway from where it crossed I-20 on up into Nashville. Depends on time, because the Parkway is a slow trip.
A whole lot depends on what you might like to see along the way. I-40 is the more "scenic" route, being the traditional tourist route from Midwest to SoCal.