You are somewhat correct. In principle the system is quantized but due to the sheer number of degrees of freedom (or quantum states) the spectra may still appear continuous.
Take for example a single water molecule in vacuum: If you calculate the QM vibrational (motion of the nuclei) and electronic frequencies you would conclude that there is no way that microwave radiation can excite a water molecule. This is because the nuclei interact with three discrete (quantized) frequencies in the infrared (IR light), whereas the electronic degrees of freedom only interact with much higher energy photons (UV and above).
However, if you now move into the bulk, you will notice that there are is also an interaction energy between the water molecules which causes rapid changes in the water conformation at room temperature. From the quantum mechanical perspective, you now get an explosion of possible states including states which excite the hydrogen bond network between the water molecules. You can now induce translation and rotation of water with respect to each other. This is why water absorbs across a large spectrum of frequencies including radiation in the microwave region.
For DNA, the situation is similar. You will also find a continuous broadening of states. BUT how well you absorb energy at a given frequency still varies. Cell phone bands do not carry enough energy to break electronic bonds directly (unlike UV light). Instead, you can only induce thermal motions of the nuclei (heat). Luckily, our DNA transfers excess heat very rapidly (~picoseconds) to its environment, so no danger of local heating to the point of bond breaking if you don't climb up a massive cell phone tower.
In the end, its very unlikely that cell phone radiation alters chemical bonds in our DNA. However, beware of the sun while talking on the phone! I would guess that outdoor usage of cell phone significantly increases the dangers of cell phone usage. ;-)