How do microwaves heat moisture-free items?

Microwave heating is largely caused by the changing electric and magnetic fields (i.e. the "microwaves") which are emitted by your microwave oven affecting polar molecules. As the direction of the electric field changes over time, the polar molecules (often, of water) attempt to follow the field by changing their orientation inside the material to line up along the field lines in an energetically favorable configuration (namely, with the positive side pointing in the same direction as the field lines). As these molecules change direction rapidly (millions of times per second at least), they gain energy - which increases the temperature of the material. This process is called dielectric heating.

However, water is not the only polar molecule in the world. You can test for yourself that most plastics don't heat in a microwave while most glass and ceramic objects do. So, a microwave oven melting your plastic bowl has more to do with it over-heating your food than over-heating that food's container.

EDIT: After doing some research to address some questions brought up in the comments to this post, I've found some very interesting information about why glass and ceramics heat up in the microwave which I will share here.

First of all, according to this article from the Royal Society of Chemistry so-called "earthenware" ceramics are fired at categorically lower temperatures than "stoneware." As a result, a non-negligible quantity of water molecules remain inside the now-seemingly-dry "earthenware," while the vast preponderance of water molecules in "stoneware" have been removed as a result of the higher firing temperature. The conclusion is that earthenware ceramics heat up in the microwave because they have the polar water molecules in them which undergo dielectric heating. On the other hand, stoneware (and apparently porcelain) will not heat in the microwave due to their respective lack of water molecules. Either way, I still wouldn't recommend microwaving your grandmother's porcelain china to find out.

Second, glass' molecular structure is apparently locally tetrahedral but without long-range order (i.e. it is an amorphous solid) which means that there tend to be spaces in the molecular structure of the glass to accommodate ionic impurities (mostly sodium, see this explanation of how glass is made to get an idea of the chemicals that go into the final product). These impurities are only loosely bound and are able to move around within the amorphous structure of the glass. These ions of sodium or other elements have a net charge (they are ions after all) which means that the oscillating electric field produced by the microwave oven causes the ions to jostle back and forth, gaining energy. The idea is very similar to the rotations of polar molecules (which have an electric dipole but no net charge), but the mechanism is different (namely, translational energy rather than rotational energy).

So in summary, ceramics apparently heat up because they still contain some water, while glass heats up mostly because of the presence of semi-free, charged ions.

This Isn't About Water

Microwave heating actually has nothing to do with the moisture content of items. It has everything to do with the amount of electric dipoles (polar molecules) in the item of concern. Water molecules (with many other organic molecules) happen to be electric dipoles. (That is, one side of the molecule has a positive charge and the other side has a negative charge.)

When the oven uses a microwave to make an electric field, all electric dipoles move to align with that field. If the direction of that field quickly flips, you give these dipoles kinetic energy. As you increase a group of molecules' kinetic energy, you increase the temperature of that group.

Any material containing significant amounts of electric dipoles will heat in a microwave. Also, resonance of water molecules have nothing to do with heating food in microwave ovens. The oscillations of the waves in microwave ovens is too slow for resonance to play a role.

For further reading, check out wikipedia's articles on Microwave Ovens and Dielectric Heating, which should answer your questions more throughly.