When I was a child we used melamine dinnerware sets and we all survived without serious health defects, so I can't help from saying that they are safe in my opinion; however, we didn't get a microwave until I was about 11 years old.
Melamine toxicity is the main safety concern with melamine dinnerware. The opinion of the U.S. Food and Drug Administration (FDA) is that food contaminated with melamine increases the risk of humans to experience kidney stones or kidney failure, which of course can lead to death. At any rate, the FDA does not disapprove of the use of melamine in packaging for food products, so obviously the concerns of it contaminating food or drink are minimal.
The FDA does issue some warnings about the use of melamine dishes. These include: (1) avoiding the use of heated highly acidic foods because the combination of acidic foods and heat can increase the amount of the chemicals that migrates from the dishes to the food; and (2) do not use melamine in the microwave. Melamine dishes are NOT microwave safe dinnerware.
Whether or not you feel that these dishes are safe for your family is really up to you. In making the decision you need to know that the chemicals used in the production of melamine dinnerware can migrate into the food, but the levels of contamination are below the comfort level of the FDA in regard to being a threat to human health, if of course you follow the previously mentioned warnings.
Personally, I am not afraid of using melamine as dinnerware. If a member of my family had kidney problems, I might feel differently about it. I think it is nice to have for outdoor dining, camping and so forth.