I'm sure the Arizona people could chime in better. Having done the first 300 miles of the AZT a Vaya would only be suitable for certain sections. The AZT is generally proper rugged mountain biking.
If you want to do some google earthing you could likely find some pretty sweet dirt roads to ramble on in Southern AZ. Again the AZ people would better help here.
Maybe a section of the ACA's Southern Tier Route:
http://www.adventurecycling.org/routes-and-maps/adventure-cycling-route-network/southern-tier/Or