Preheat oven to 375°F (190°C) and lightly grease a 9x13 inch baking dish.
Add the cubed chicken, uncooked rice, chicken broth, minced garlic, dried oregano, salt, and black pepper directly into the baking dish and stir to distribute ingredients evenly.
Cover the dish tightly with aluminum foil to prevent steam from escaping.
Bake for 40 to 45 minutes, or until the liquid is fully absorbed and the chicken reaches an internal temperature of 165°F (74°C).
While the dish bakes, whisk together the Greek yogurt, grated cucumber, lemon juice, and half of the fresh dill in a small bowl to prepare the tzatziki sauce.
Remove the baking dish from the oven and let it rest, covered, for 5 minutes.
Remove foil, fluff the rice with a fork, and top with dollops of tzatziki sauce, crumbled feta cheese, diced red onion, and the remaining fresh dill.