Microwaves are almost certainly safe, but just FYI, the point about there being 'no plausible mechanism' is wrong, and a common misconception. Photons don't need to have enough energy to directly cause DNA breakage, in order to be dangerous. Microwaves seem have effects on proteins beyond that caused by thermal excitation, which means they could plausibly be carcinogenic, e.g. if they interfere with DNA repair enzymes. There's some evidence that pumping enough microwaves at cells in culture can turn them cancerous.

The epidemiological evidence though, is that they don't cause cancer.

http://www.ncbi.nlm.nih.gov/pubmed/11088227