Trim and discard excess fat from meat (dripping fat can cause flare-ups). Rinse pieces and pat dry; if necessary, cut into serving-size pieces.
Place meat in a heavy zip-lock bag (1-gal. size; see notes). Seal bag and turn to coat pieces in marinade. Chill, turning occasionally, at least 30 minutes or up to 1 day for meats and poultry, 20 to 30 minutes for fish.
With tongs, lift pieces from bag and lay on a barbecue grill 4 to 6 inches above a single, solid layer of hot coals or high heat on a heated gas grill (you can hold your hand at grill level only 2 to 3 seconds; see notes); close lid on gas barbecue. Discard marinade.
With a wide spatula or tongs, turn pieces over halfway through cooking. (For fish fillets with skin, grill skin side down first; to turn, slip spatula under flesh and flip onto another place on grill. Remove and discard skin.) Cook beef or lamb until done to your liking (cut to test), 8 to 10 minutes total for medium-rare; pork and chicken until no longer pink in center of thickest part (cut to test), 9 to 12 minutes total; or fish until barely opaque but still moist-looking in center of thickest part (cut to test), 9 to 12 minutes total. Transfer meat to a board or platter and let rest 2 to 3 minutes before serving.