Mix cider vinegar, mustard, garlic, lime juice, lemon juice, brown sugar, salt, and pepper together in a large glass or ceramic bowl. Whisk in olive oil. Add chicken and toss evenly to coat. Cover, and marinate in the refrigerator for 8 hours to overnight.

Preheat an outdoor grill for high heat.

Lightly oil the grill grate. Place chicken on the preheated grill, and cook 6 to 8 minutes per side, until juices run clear. Discard any remaining marinade.