I've biked across Nebraska many times - had a great time.
I driven practically every backroad - the small towns are great!
No matter where you cross in the Great Plains in mid-summer - it will be hot.
Just look at a number of journals about Kansas on the TransAm over at Crazyguyonabike.
They have a traffic volume map at the Nebraska Department of Roads website. Unfortunately it is down this weekend so I can't give you the link. Look at the variations in traffic - esp. in the east. There's also a new rail trail called the Cowboy Trail that stretches from Norfolk to Valentine. Check it out:
http://www.ngpc.state.ne.us/parks/gu...boy/cowboy.asp
A number of people have commented that it was actually easier to ride on the highway - I don't know. I have biked NE 2, US 20, US 30, and NE 92 in the the west; NE 74, NE 4, and NE 2 in the east - plus a couple of more that I can't list off the top of my head. US 20 has more traffic and more trucks than NE 2. NE 2 has a railroad that generally follows it with frequent coal trails breaking the quiet. Yes, western Nebraska is largely sandhills. I find them beautiful. Shade is at a premium - but that's why they call the region the Great Plains - it's a grassland.
Depending where you are going to be in eastern Nebraska, I'd avoid NE 2. NE 4 is very nice in southeast Nebraska. If you are staying further north and using US 20 - make sure to swing over to NE 12 - it's a fabulous ride.
Most of the little towns have community parks where you can camp. The motels - where there ARE motels - may still be $30. I've never had trouble one in Nebraska - people have always helped me and been very generous.
E-mail me and I'll answer any questions you have directly.
Best - J
PS - Pic is of the Loup River in central Nebraska