"A microwave oven cooks food because the water molecules inside it absorb the microwave radiation and thereby heat up and heat the surrounding food. Microwave radiation will similarly heat up skin and other body parts. In fact, people stationed at big microwave towers in cold climates used to stand in front of the microwave generators in order to warm themselves. The radiation is harmful mostly to the parts of the body that cannot conduct the heat away very effectively---the eyes especially. I think that heat transfer could explain why one sometimes hears about people (fast-food workers, for instance) getting headaches when exposed to leaking microwave ovens."
David E. Hintenlang, associate professor nuclear and radiological engineering at the University of Florida at Gainesville, adds some further details:
"Microwave ovens cook food by generating intermolecular friction between the molecules of the food. The microwaves cause water molecules to vibrate; the increased friction between the molecules results in heat. Microwaves could affect your tissue in a similar way if they were able to escape from the microwave oven. Modern microwave ovens are designed to allow essentially no leakage of microwaves, however. The only time for concern would be if the door is broken or damaged, in which case the oven should not be used.
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