I knew this would be a problem:

The Unintended Consequences Of A Gluten-Free Diet, Epidemiology, online 15 March 2017

Reseachers found higher levels of arsenic and mercury in the urine and blood, respectively, of participants eating a gluten-free diet compared to non-gluten-free controls.

To our knowledge, this is the first analysis to suggest that Americans on gluten-free diets may be exposed to higher levels of arsenic and mercury.

Why?

Rice may be contributing to the observed higher concentrations of metal biomarkers among those on a gluten-free diet as the primary substitute grain in gluten-free products.

This is from Howard G. Smith MD at Business Insider:

Studies have shown that rice accumulates these toxic metals from fertilizers, pesticides, the soil, the rocks and the water from which it is grown.

Although it is a deadly poison in huge quantities, arsenic in small quantities can still increase the risk of bladder, lung, and skin cancers, as well as heart disease and Type 2 diabetes. Mercury, on the other hand, produces neurological, psychological, and hearing issues.

Air pollution is also contributing. Coal-fired power plants are the biggest industrial emitters of arsenic and mercury … which fall onto soil and into the flooded paddies in which rice is grown. Even organic rice has high levels of arsenic. Fixing this problem means cleaning the air, which means regulating power plants, which is done in part by the EPA, which is an agency being defunded by the current administration.

Related (Thanks to Shaun):
Low Gluten Diets May Be Associated With Higher Risk Of Type 2 Diabetes, American Heart Association Meeting, Presentation 11, 9 March 2017