In a medium bowl, whisk together 2 tablespoons olive oil, 2 tablespoons lemon juice, minced garlic, 1 tablespoon dill, oregano, salt, and pepper. Add chicken breasts and turn to coat. Cover and refrigerate for at least 15 minutes (or up to 2 hours).
In a small bowl, combine Greek yogurt, 2 tablespoons lemon juice, 1 tablespoon olive oil, 1 tablespoon dill, minced garlic, salt, and pepper. Stir until smooth. Cover and refrigerate until ready to serve.
Preheat a grill or grill pan to medium-high heat (about 400°F). Remove chicken from marinade and discard excess. Grill chicken for 5-7 minutes per side, until internal temperature reaches 165°F and grill marks appear. Let rest for 5 minutes, then slice against the grain.
Divide the cooked rice evenly among 4 serving bowls. Arrange cucumber, cherry tomatoes, red onion, and feta cheese (if using) in sections on top of the rice.
Top each bowl with sliced grilled chicken. Drizzle generously with lemon dill sauce. Garnish with fresh dill and serve with lemon wedges on the side.