Sprinkle the garlic and salt in a shallow dish. Mash the garlic with the blade of a knife. In a small bowl, combine the mashed garlic, parsley, tomatoes, and oil. Mix well. Cover and refrigerate for 8 hours or overnight.
Prepare the grill for cooking. Cut two lengthwise slits in the salmon fillet with a sharp knife, dividing the surface of the fish into thirds. (Cut to the skin but not through it.) Spread half the garlic mixture over the fillet and into the slits. Place the salmon, skin side down, on a greased grill rack. Cover the grill and cook over low heat for 10 to 15 minutes. Spread the remaining garlic mixture on the fish. Continue cooking, with the lid down, over medium heat for 15 minutes, or until the fish flakes easily. Remove from the grill by inserting spatulas between the skin and the flesh, lifting the fillet, and leaving the skin on the grill. Serve the skinless, boneless fillet on a bed of fresh greens.