They do (well sorta), its a common misconception that microwaves only heat water molecules. Here's how a microwave works. Many (most) molecules have what is known as a dipole moment, molecules with such a moment are caled 'polar'. What this means is that there is a non-uniform charge on the molecule. Take for instance the molecule A-B. A pulls electrons to it stronger than B does, which makes A more negative and B more positive, which makes it polar.
Polar molecules will align themselves in an electromagnetic field. A microwave oven generates that field. By alternating the field the angle of alignment is changed, which speeds up the rotation. To make a comparison, think of a merry go round on the playground. Your friend gets on it (drunk of course) and you grab a handle and spin it. Each time it goes around you grab a handle and spin it again. After a couple goes your friend is spinning really fast and performing a move that I like to call 'the vomit sprinkler'.
Heat is, by definition, kinetic energy. Rotational motion of a molecule is a form of kinetic energy. Therefore, by speeding up the rotation of the molecule you increase the heat of the molecule, and therefore the overall heat of the system.
So, for a microwave to heat something, you just need a polar molecule. Many many many many molecules found in organic systems are polar, so you end up having all sorts of stuff getting hot, which hopefully will make the whole system uniformly hot.
Here's where its different from conventional heating. In a microwave all of the energy is coming into the system at the polar molecules. This means that it isn't uniform, which means that an area with a higher concentration of polar molecules will reach a higher temperature than the system as a whole finishes at after the heat transfers. The transfer of energy happens fast, but in some systems it could leave you with superheated regions.
An example of this is the experiment where you put a tall container of water in the microwave and turn it on for a while. When you pull it out its not boiling, but if you dip a spoon into it the water flash boils and explodes all over the place (by the way don't try this at home, its seriously dangerous). This is because the water in the bottom of the container is well above the boiling point, but isn't boiling due to the pressure of the cooler water above it. In lab work this is what's known as 'bumping' and it is really bad (I once had a vial of benzoic acid bump all over my face.) To avoid that people use 'boiling chips' which are high surface area, inert pebbles that induce boiling (which causes enough convection to maintain an even temperature.)
Anyways, totally digressing. The point is that in food that is being microwaved you could have very small superheated regions that are undergoing reactions that would not be possible at normal (or cooking) temperatures. You may think its crazy to think that you are getting to that high of a temperature in a reall tiny area, but its not something unique to microwaves. There is a tool called a(n) (ultra)sonicator that basically just shakes really fast, and it can get local (and really small) temperatures up to 5000 K, slightly less than the temperature of the sun, 5800K (this is actually caused by creating miniature vacuums/cavitation, not from direct heat generation from the shaking.)
Anyways, it is possible that in those microscopic (really nanoscopic) high temperature regions you are undergoing all sorts of crazy reactions that are energetically impossible at lower temperatures, and with the vast array of reactants you have in an complex biological medium (aka 'food') you could end up with some very toxic substances in it.
HOWEVER. The mitigating factor here is scale. You may be generating a toxic molecule, but you may literally only be generating a single or a handful (which could be thousands and its still no biggie) of molecules, so it probably doesn't matter. But the mechanisms are there.
