In a medium saucepan, combine Dr. Pepper and next 8 ingredients. Bring to a boil, reduce heat and simmer, 30 minutes, stirring occasionally. Let cool.
Place chicken in a shallow dish. Pour 1/2 of sauce over chicken, and marinate 30 minutes, turning occasionally. Grill chicken, covered with grill lid, over medium coals about 35 minutes, or until meat thermometer inserted in thickest part registers 180, turning occasionally. Brush chicken with remaining sauce while cooking.