Having grown up in Järna, which may be the most pseudoscience-dense place in Sweden I have learned that microwave ovens are dangerous and that it kills your food (yes that is the term you normally hear).
Their arguments, of course, rely heavily on the naturalistic fallacy i.e. they claim that heating using microwaves is somehow "unnatural", that is unlike conventional oven which they seem to consider "natural". It is as if electromagnetic radiation in the infrared spectrum is somehow more natural than electromagnetic radiation with microwave frequencies.
There are however also some more empirical claims used by people who are against microwaves. For example many people quote a "study" where they compared growth of plants which had received either microwaved water or water which had not been heated. It is claimed that the plants which got the microwaved water waned down and died. The plants which had just gotten normal water on the other hand flourished. According to microwave critics this shows that microwaving water takes out some sort of "life energy" out of the H2O molecules and therefore the plants die.
Another empirical claim that microwave antagonists sometimes use is that microwaves are forbidden in Russia because they are so dangerous – this is just a simple lie.
One claim with some actual research behind it is that microwave ovens destroy certain vitamins in your food (see here for example). This claim is probably true but only if you compare microwaved food with raw food. The study quoted on health bulletin shows that meat heated in microwave ovens has 30-40% less vitamin B12 than raw meat. So if you start eating your steaks raw you can skip a glass of milk a day (which would otherwise compensate the loss).
"Microwave ovens are widely used for cooking and food processing. Extensive studies (Cross and Fung, 1982; Hoffman and Zabik, 1985) have shown equal or better retention of some vitamins (B1, B2, B6, C, and folic acid) after microwave heating compared with conventional heating."
In other words, if you lack vitamins, cooking in your microwave can help you.
It is also interesting that otherwise environmentally conscious people are so against microwave ovens even though they save a lot of energy compared to conventional ovens.
So where does all this leave us? My conclusion is that microwave heating is a fast and convenient way of cooking. Don't put eggs or metal in there and don't overheat things so that you burn yourself. If you follow these guidelines a microwave oven will be a terrific assistant in your kitchen. Of course some things just taste better when you cook them in a conventional oven and in that case I am all for that…
If you want more on this topic I can recommend this episode on Brian Dunning's skeptoid.