For allergy sufferers, there is simply no safe haven. While desert regions have no maple trees or ragweed, they do have plenty of other plants that produce pollen, including sagebrush and cottonwood, ash, and olive trees. Relocating to such a region may offer relief for a few months, but a fresh crop of allergies to local plants is likely to develop before long.
Moving from one city to another can actually trigger seasonal allergies in many people. HowStuffWorks explains the details behind why.