We pedaled a couple thousand miles through deserts with our kids on our last tour, and rarely had problems. As others have mentioned, be sure to take plenty of water, but you don't have to be fanatical about it unless you are crossing a long stretch where there is nothing - we did a 93-mile stretch with nothing, but some cars stopped and gave us water so we were fine. I ran out of water once and had to hightail it to the next place I could get some.

We found we didn't get many flats as long as we only rode on the roads. If we were getting off the road for any reason (we go back and camp in the desert each night, as well as eat lunch off the road) we walked our bikes. As soon as we got back on the road, we did a thorough check of our tires to pull out any thorns before we put our weight on the bike.

If it's really hot (think 95 degrees or more) then you'll have to take special precautions such as the salts and all that - but most of the time just be logical in your efforts.

Have fun!