I lived in Phoenix for a couple of years and here are a some of my personal favorites.
Richardson's isn't too far from downtown and a real gem. They do a very flavorful version of New Mexican cuisine. Very good eats.
http://www.burningembersphoenix.com/Chino Banditos Takee Outee is out on 19th Ave between Thunderbird and Bell across from Turf Paradise. Its a pretty good distance from Downtown but what really isn't. Phoenix is the textbook example of urban sprawl. Anyway it is well worth the trip and really cheap as well. You may want figure out how to order by looking on the website first because it can be a little confusing and the lines can be long.
http://chinobandido.com/The only place I liked Downtown is Alice Cooperstown on Jackson down by the arena and baseball park. Its a cross between A Hard Rock Cafe and a ESPN Cafe with some pretty decent BBQ.
The waitstaff used to wear Alice's eye makeup. Pretty cool place.
http://www.alicecooperstown.com/index.htmlDurant's on Central is a real old school steakhouse in the Rat Pack tradition. The food is pretty good but the atmosphere really makes the place what it is.
http://www.durantsaz.com/Last but not least. The 5 and Diner on 16th Street is a real, honest to goodness 24 hr. stainless steel diner with darn good eats. I suggest the Chorizo Omelet or the Blue Moon Burger.
http://www.5anddiner.com/index.html