You can often do so, especially with indoor electric barbecues. Outdoor models, on the other hand, are a different story. It is, however, dependent on the materials and/or nonstick coatings used. The only safe solution is to follow the manufacturer's directions. It's a good idea to first rinse out the worst of the residue from those pieces that can go in the dishwasher, so it doesn't clog your machine's filters.