Microwave ovens are able to cook food because they deliver so much energy via microwaves that the molecules of the food vibrate, producing heat. Standing right next to an old microwave oven is "only" dangerous because the microwaves could burn you. Temperature changes in certain parts of the body can lead to problems (like temporary infertility), but not cancer.
The only kinds of electromagnetic radiation that cause cancer are the ionizing ones, like gamma rays. Fortunately, microwaves are not ionizing - not even close. Normal visible light is closer to gamma rays on the EM spectrum than microwaves are. Similarly, the electric field from power lines is non-ionizing, and so cannot cause cancer.
"Let's see some reliable sources!" Sure:
- Microwaves, Radio Waves, and Other Types of Radiofrequency Radiation from the American Cancer Society
- Power Lines, Electrical Devices and Extremely Low Frequency Radiation likewise
- Cell Phones and Cancer Risk from the National Cancer Institute
- Electromagnetic Fields and Cancer likewise
Note that studies are still going on; there is yet some uncertainty. So far, the evidence for a causal relationship between microwaves/phones and cancer is not there.