I just recently did this route.
I left from Duluth, MN with a goal of Tucson, AZ
Duluth to Pipestone (there's a great old hotel, the Calumet Inn, in Pipestone and there's also excellent pie at the local diner, Lange's Cafe)
I then took Hwy 81 and stayed in Concordia, KS because it started pouring down rain (stayed at a Super8 there - nothing to write home about but was quite nice). At Salina, KS, I took 140 to Elsworth and then 156 to Great Bend. 156 becomes 56 and takes you to Clayton, NM
In Clayton, NM stay at the Eklund Hotel and check out the hardware store across the street - the old lady there is 98 and has worked at the store since she was a kid - very cool. There's also a great NAPA auto parts store that a young woman recently opened and she's quite interesting to talk to, too.
For a scenic side bar, from Clayton, head out on 87 instead of 56 and check out the Capulin Volcano. Then head on down through Santa Fe and Albuquerque. At ABQ, hang a right and head west towards Grants, NM and then take 117 down through El Malpais National Monument. You'll then find yourself in Quemado, NM where you can detour east to PieTown on Hwy 60 and get some more really good pie or continue southwards towards Silver City.
Anyway, this is a really nice way to get from MN to AZ without very little interstate and really nice roads.
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