In a small bowl, mix together brown sugar, honey, liquid smoke and vinegar.
Brush one side of the salmon with the basting sauce. Place the salmon on the grill, basted side down. After about 6 minutes, generously baste the top and turn over. Cook for about 7 more minutes, then brush on more basting sauce. Turn and cook for 2 more minutes. Adjust cooking time if using thin salmon fillets. (Take care not to overcook the salmon as it will lose its juices and flavor if cooked too long.).