The problem with using a dishwasher to clean the grilles isn't the grease, it's the carbon black, which is completely insoluble. It will just gradually flake off and get the rest of your dishes dirty.
A gas grill is your friend. If I leave mine on high for 15 minutes it gets to over 750F, which is the same temperature as a self-cleaning oven. This vaporizes everything, including the carbon black, which just oxidizes to CO2. When my grill gets really crusty I just run it on high for 45 minutes and all that's left is a coating of ash that blows right off.
The only caveat is that my grill racks are thick, high-quality stainless (Weber Summit). I don't know how the CS plated racks would handle the 750F treatment. Has anyone tried?