The notion that they are safe is based on ionization. Cell phones emit non-ionizing radiation. This is critically different than the ionizing radiation you find accompanying for example: x-rays and nuclear waste. Essentially it means that the radiation radiating out of your cell phone isn't strong enough to ionize electrons or particles in atoms. It is through ionization of atomic particles that we beleive radiation causes genetic damage; the mechanism by which it causes cancer. This exceedingly grim possibility however remote is what lends the question to further scrutiny. The ongoing question now is whether or not long-term cell phone use adds additional risk. Scientific American summarized the bad news:
"Three studies since 1999 indicate that people who have used cell phones for more than a decade may have as much as three times greater risk of developing brain tumors on the side of the head against which they most often hold their phone"Current regulations only address short term concerns and more immediate exposure risks. The exposure standard for these devices is quantified by it's Specific Absorption Rate (SAR). the current SAR limit is 1.6 watts per kilogram averaged over one gram of tissue. That standard is accepted by the FCC and Industry Canada. But it fails a bit for me. there are three types of radiation and it rates all three equally. That's just bad science. Not all radiation is created equal. The SAR measures fields between 100 kHz and 10 GHz. While that does envelop all fourteen GSM bands, it doesn't weight them by their relative ability to ionize. Overall the power is low and I don't expect that any cellphones are particularly dangerous especially compared to the known sources of ionizing radiation ex. the sun.
here, here and here.