Interesting timing: I just spent a week in the desert on the California-Arizona border.
You didn't mention time of year or day, which are big factors. If there are any flowers in bloom (spring), bees will be everywhere. Mornings will definitely have birdsong. Depends on the altitude, too, and terrain; there'll be some life everywhere unless, as stated before, it's dunes, but that's rare terrain. Frogs and crickets you'll hear mostly at night, near water and non-succulent plants, respectively, both of which can be scarce. (Make no mistake, the smallest water-filled hole can scream with frogsong after dark.) Raptors inhabit both open and mountainous desert terrain, including owls, and ravens are also common (stay away from that DAMN HAWK! :-p). Coyotes at night for sure. Lizards will often make all sorts of skittery movement sounds as humans (or any threats) approach. And, of course, wind in the bushes, and the scouring of wind-borne sand when things get really blustery (especially across roads).
Don't forget to keep altitude in mind; Arizona has quite a lot of variation in the height of its terrain!