If you have to eat in Kingman, make it the In And Out that just opened at Stockton Hill Rd.
Grants, NM is home of 4B's, better than average truckstop fare, KILLER tomato soup.
Albuquerque offers plenty of choices. Blake's Lotaburger is a local statewide chain that does an excellent green chile cheeseburger by fastfood standards. For upscale New Mex Mex there's Sadie's. The Frontier might be right up your alley...ABQ's quirky 24-hour joint.
Try to hit Amarillo for breakfast at the The Big Texan steak house. Awesome steaks, and they are about 40% less at breakfast time.
Once you leave I-40, you'll go through Wichita Falls to get to Dallas. The Pioneer is a local chain of diner style places that has been solid for years.
re: Jerry Saywell
I strongly second the Turquoise Room at La Posada. We always schedule it for breakfast, when near Winslow - even coordinating travel plans around this.
In Holbrook, I like Mr Miestas' on Route 77, on the southern edge of town (east side of highway, just before you leave town). Great AZ/Mex lunch. Too bad they are so close to Winslow though. Too much food, and not enough miles!