In a small bowl, mix paprika, thyme, salt, and pepper. Rinse ribs and pat dry. Rub herb mixture over both sides of each rack, pressing so it sticks. Wrap each rack in heavy-duty foil (see Notes).
Prepare grill for indirect heat: If using charcoal, ignite 60 briquets in a chimney starter (or mounded directly on the firegrate). When they're dotted with gray ash, in 15 to 20 minutes, push equal amounts to opposite sides of grate. Add 3 more briquets to each mound now and every 30 minutes while cooking. Set a drip pan on grate between mounds, then set cooking grate in place. If using gas, turn all burners to high, close lid, and heat for 10 minutes. Then turn center burner(s) off so heat is at edges of grill, not under cooking area; turn outside burners to medium.
Lay foil-wrapped ribs on grill, convex (meaty) side up, not directly over heat; overlap slightly if necessary. Cover and cook until tender when pierced (through foil), 1 to 1 1/4 hours. Carefully remove foil from ribs. Brush tops (meaty side) lightly with cherry-Zinfandel barbecue sauce, turn over, and cook until sauce is browned, about 10 minutes. Brush concave (bone) sides, turn again, and cook until browned on that side, about 10 minutes longer.
Transfer ribs to a board and cut between bones into individual ribs. Season to taste with salt and serve with remaining barbecue sauce.
Note: Nutritional analysis is per serving.
Wine pairing: Zinfandel