Prairie Dogs. They're super tough, smart, hard to find and stalk, and take a lot of killing, plus you get a lot of great, delicious roasts and steaks. The trophy Prairie Dogs also make fantastic head mounts.
I use a .416 Rigby, as I don't want to lose an animal. Puts 'em down right quick!
Having never hunted and harvested any big game animal besides whitetails, I'll have to go with that for now. However, I did see how hunting the open spaces in AZ could be addicting. I plan on broadening my horizons quite a bit this year!!!