I think a lot of this is driven by technophobia. The enigmatic "they" made similar claims about electric lights in the home (disproven), the television and sitting too close (disproven), radar range/microwave ovens (disproven), cordless telephones (disproven), computer CRT monitors (not yet disproven, but few statistically significant cases can be found), and now the ubiquitous cell phone.
Cell phones have been around for over 20 years, and have had an almost exponentially increasing market share for about the past 12 years. If there were any statistically significant rates of brain cancer (or other major adverse health effects) amoung cell phone users, I'm sure some lawyer would have filled a class action suit by now.