For the marinade, combine all ingredients except the chicken in a gallon sized ziploc bag. Add the chicken, seal the bag, and toss to coat. Refrigerate for at least 2 hours, up to 24 hours.
For the beets, bring a large pot of water to a boil. Add beets, and boil for 10-12 minutes, or until beet cubes are tender when pierced with a fork. Strain and run cold water over the beets to cool them.
Preheat your grill over medium high heat.
Once beets are cool, toss them with olive oil, salt, and pepper. Add them to a grill basket.
Place the grill basket and marinated chicken legs to your preheated grill. Reduce grill heat to medium under the beets and medium low under the chicken. Cover. Let cook for 7 minutes, then flip the chicken and toss the beets. Cook for another 7 minutes then remove chicken to a plate and remove grill basket.
While the chicken and beets are cooking, prepare the coconut lime sauce by adding all ingredients to a small sauce pan. Cook for about 5 minutes over low heat, or until smooth and warmed. Taste and adjust salt and lime juice as necessary.
Serve over steamed white rice (optional) with coconut lime sauce poured over the chicken and beets.