In a large bowl, combine olive oil, lemon juice, minced garlic, oregano, salt, and black pepper. Add the chicken pieces and toss to coat. Cover and marinate in the refrigerator for at least 30 minutes, or up to 2 hours for more flavor.
While the chicken is marinating, prepare the lemon rice. In a medium saucepan, bring chicken broth to a boil. Add the rice, lemon zest, and lemon juice. Stir, reduce heat to low, cover, and simmer for about 15-20 minutes, or until the rice is tender and the liquid is absorbed. Remove from heat and let it sit covered for an additional 5 minutes, then fluff with a fork.
Preheat your grill or grill pan over medium-high heat. Thread the marinated chicken pieces onto skewers, leaving a little space between each piece.
Grill the skewers for about 10-12 minutes, turning occasionally, until the chicken is cooked through and has nice grill marks.
Serve the chicken skewers over the lemon rice and garnish with fresh parsley.