While I'm still thinking about it, since you are going to Tucson, the quickest route is probably the southern route, bypassing Oklahoma and/or Kansas, and most of Missouri. It is one we've taken to get to southern Texas in the least amount of time.
I-69 to I-94 to I-80 to I-57 to I-55 (Sikeston) to I-40 (West Memphis) to I-30 (Little Rock) to I-20 (Dallas/Fort Worth) to I-10 (somewhere west of Pecos). This route makes maximum use of diagonals to shorten the distance, and is almost all flatland (except for some hills in western Arkansas) until you start climbing over the Rockies. At DFW, I would take the I-635 loop around the east side of Dallas to get to I-20 early, rather than I-30 through the Metroplex to the junction west of Fort Worth. I-30 gets seriously tied up with suburban commuter traffic, I-20 is just very busy rural Interstate.
Other than Dallas, the only real hot spots on this are the mess where I-55/40 run together through West Memphis and seems to be permanently under reconstruction, and of course I-80/94 from where they join east of Gary to where I-294 pulls the Chicago traffic away. I've gone through that in less than 25 minutes, and other times I've spent more than a hour creeping through. I have alternative routes from I-94 out of Michigan to I-57 in northwestern Illinois, but they are much slower than staying on the Interstates, if the Interstate are flowing normally.
