It is an interesting and constantly recurring question that you ask. The body of scientific evidence that supports any negative health effects from well placed and maintained mercury amalgam fillings does not exist.
There is, however, plenty of evidence that mercury amalgam fillings produce a raised level of mercury in the mouth when they are placed, polished or removed and also to a lesser extent when one has a hot drink or acidic food and drink, this can be reflected in trace form more centrally in the body.
Some individuals may be more sensitive to mercury amalgam fillings than others and this can be seen by innocuous gum lesions adjacent to large fillings in the same way that some people react to Nickel in earrings for example.
There is no proven link with mercury amalgam fillings to major health problems and illnesses such as ME and multiple sclerosis although these linkages have been claimed by some practitioners, but these claims have been denigrated by most scientists and clinicians
The above is a summary of the main debate and evidence thus far. Amalgam fillings have the greatest longevity of all fillings directly placed in the mouth as opposed to those constructed of gold or porcelain in the dental laboratory; are generally well tolerated by teeth and appear to be stable, preventative of recurrent tooth decay and relatively simple to place by all levels of dental practitioners. They are however unsightly and waste amalgam from drilled out fillings may have some detrimental impact on the environment if not disposed of correctly.
Whether one should have amalgam fillings as opposed to an alternative filling material therefore comes down to personal choice as a consumer, the particular site the filling is to be placed in and the relative ability of the dental practitioner completing the treatment.