Your friend is, very, very theoretically, right, but the risks on both theoretical grounds and also epidemiological grounds - i.e. microwave ovens have been used by many people for a long time without obvious illnesses showing themselves - are extremely small.
There are two ways wherein microwave cooking might "change the molecules": the first
They might break and reconfigure bonds within organic molecules. However, whilst this theoretically happens, it happens unbelievable seldom if practically at all. Bond energies and bond dissociation energies are of the order of electron volts or tens thereof. So they are a few or a few tens of optical photons' worth of energy: bond reconfiguration is thus driven by photons with frequency of the order 1000THz. Microwave oven photons, on the other hand, at 1 to 2 gigahertz, are six orders of magnitude less energetic. However, from quantum mechanics, there is a nonzero probability that bond breaking by microwave photons will happen, but it will be fantastically low. This is the idea of quantum tunnelling: if and event, through energy considerations, is forbidden classically, it still happens, albeit seldom. Cold hydrogen fusion happens, for example, when you pull sticky-tape off something, but the events are fantastically seldom.
Microwaves denature proteins through their pure heating effect, i.e. change their three dimensional shape without changing the chemical bonds within them. An analogy is supercoiling and curliness in a telephone receiver cable. The basic cable can stay intact, but different amounts of winding can get it "stuck" in configurations of different 3D shape (like the kind where it's supercoiled so much the knots wrap themselves around your hand when you're trying to talk on the telephone and your interlocutor, if unlucky, thinks they're getting sworn at). However, this denaturing is exactly the same effect as wrought by any other kind of heating. Protein denaturing is essentially the difference between cooked food and raw, whatever the heat source used for the cooking was.
So yes, the molecules do change, but in ways that are pretty much the same as changes wrought by any kind of heating, or even folding (as with an egg white - the whitening of whipped egg is owing to mehcanically wrought denaturing).
This article here is a more learned exposition on some of my ideas above.
Edit After Interesting Comment:
User Davidmh made the following comment on Volker's Answer:
Recipe: potatoes sliced in the microwave. Some of them, the ones in contact with the container can get very toasted, as if you grilled them.
This raises an interesting point. Although I believe the potato toasting is still a pure heating effect, there may indeed be an effect at work here that's peculiar to microwave cooking. The food in the microwave is interacting with the electomagnetic radiation, and so there must be a reaction - or scattered - electromagnetic field so that the food changes the field distribution within the resonant cavity. What you're seeing here is probably a combination of all four of the following:
- Microwaves are excluded more from deep within a big mass of food than they are from the edges: the changed field configuration means the edges get more heat. This is a bit like the electromagnetic skin effect;
- Near an edge, the food is less well "heat sink-ed". Heat can diffuse off in all directions away from a locally hot region within the body, so the heating tends to be made even within the body of a mass of food. At the edges, there are fewer diffusive paths for the heat to get away from locally hot regions;
- Water boils off in a microwave oven. It cannot get away from the inside of a mass of food, and so it tends to set up water-steam equilibriums inside the food body and thus tend to keep the temperature nearer to 100C. But water can boil off the surface of a food body. So the edges tend to dry out swiftly and, lacking the liquid water-steam equilibrium that tends to constrain the temperature, can rise to a much higher temperature.
- If the container is at all conductive, it will absorb the microwaves and become a local hot spot. I have found some earthenware dishes and pots do tend to get extremely hot on their own in a microwave oven. One of the factors in declaring something "microwave safe" is whether it absorbs in this way: not only does it heat the food unevenly if it does, it can destroy itself.
You could test how much 4. is a factor with a particular pot by putting it into the microwave with nothing in it and seeing whether it heats. BTW make sure you switch the microwave on for the test: I was trying to debug a test setup a few days ago and took two hours to twig that I hadn't switched the power on to a key piece of kit!