Can stainless grill grids go into the dishwasher?
Sure, but Why?
The whole idea of grilling is easy clean up. Just burn them off prior to the next use, and give them a quick brush with a brass grill brush.
If you have cast iron grates I have found it useful to keep them coated with cooking oil just like you would season a cast iron skillet...even some of the cheaper grills stainless is prone to corrosion, and can benefit from a coating of cooking oil.
Oil cleans oil. Grease cleans grease! I use my old rancid olive oil on the grids and brush with the metal grill brush, and EASY clean up!!