I lived in Prescott for several years. If you're going to be cycling much above 5000' in central or northern AZ, be prepared for several days of winter weather in March and early April. (Up on the Colorado Plateau, I was in a post-Easter storm with temps in the teens, extremely high winds, and several inches of snow--horrible cycling. Luckily I was on a hiking trip and could keep going.) Keep an eye on the forecast and head below 4000'--plenty of fine cycling down low.

I'm in the "forget about staying dry, get wet and stay warm" school. After a couple of decades of many hundreds of dollars spent on breathable rain gear, I no longer use it. The last ten years or so (five of those in Arizona) I've switched to silnylon, extremely lightweight (7 oz for anorak and pants), packs to the size of a fist, and cheap (homemade by a friend). I used bread bags for my feet the few hours I needed it over the years I lived in Arizona--free, extremely small and light. I pay more attention to the layers underneath, ventilating, and drying out as often and as soon as possible--trading off experience for expensive gear.

If you have the budget for it, the expensive stuff can be very nice in many conditions, but it's not absolutely necessary. (Keep in mind if the membrane gets wet, from inside or out, it stops breathing.) You will be in a southern desert after all and you might not need it at all with a little luck.