Actually, the fact that they are non-ionizing doesn't prevent them from harming DNA. Ionization loosely means that the power is sufficient to destroy a base pair in a DNA chain (via striping of an electron), if the full energy of the wave packet is absorbed. Ionizing radiation is guaranteed to hurt you if it is absorbed by your body (e.g., it will ionize something whether that is protein or DNA). My perception of why "non-ionizing" doesn't mean it is safe comes from a (tangential) education in terahertz radiation (e.g., microwaves). Simply put, just because the radiation may be low in power when averaged over time and space, the instantaneous energy density of the radiation might make it unsafe. DNA can be harmed through lots of different ways other than ionization (strand separation, mutagens, denaturing, etc.)
For an ocean analogy, just because the ocean has an RMS wave height of 5 feet doesn't mean that *all* the waves will be 5 ft tall. Instantaneous peaks (in space and time) will discharge sufficient energy (albeit non-ionizing) into DNA to cause the strands to separate (and be subject to other effects accordingly). For a gadget example, take the microwave. It isn't ionizing. It doesn't directly cause cancer, but if an organism is subjected to sufficient microwaves of power to denature proteins, the process will cause upticks in cellular metabolism to repair those proteins. I for one do believe that the uptick in metabolism does in fact lead to a higher incidence of cancer (though metabolic studies vs cancer rates are really not well documented in my book and mostly involve healthy people starving themselves).
I think the best take on cell phone radiation, for which sadly cannot attribute, was from a UK doc several years ago who was worried that the digitization of cell phone signals (vs analogue), while it would lead to a much lower RMS would also lead to bursts of *very* high instantaneous energy. This might denature proteins over time, like cooking an egg millimeter by random millimeter.
Forget studies on people with cell phones for the next decade or so. People are complicated and are difficult to pin down w.r.t. a cause of a disease. I think we probably need to spend more money on actual fundamental (microbial) research on non-ionizing radiations effect on cellular growth (such as http://www.biomedsearch.com/nih/Cell-phone-radiations-affect-early/20355324.html). As for myself, right now I have no idea if they are safe, but I for one know that just being "non-ionizing" isn't enough.