In medium nonmetal bowl, mix relish ingredients. Cover; refrigerate at least 1 hour to blend flavors.
Mix chili powder and paprika; sprinkle over chicken.
Heat gas or charcoal grill. Place chicken, skin sides up, on grill rack over medium heat. Cover grill; cook 15 minutes. Turn chicken. Cover grill; cook 20 to 40 minutes longer, turning occasionally, until juice of chicken is clear when thickest piece is cut to bone (170°F for breasts; 180°F for thighs and drumsticks). Serve with relish.