In a medium bowl, whisk together olive oil, lemon juice, dill, garlic, 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 saucepan, combine rice, water, and salt. Bring to a boil, then reduce heat to low, cover, and simmer for 15 minutes. Remove from heat and let stand covered for 5 minutes. Fluff with a fork and stir in butter if using.
In a small bowl, combine Greek yogurt, lemon juice, dill, garlic, and salt. Stir until smooth. Refrigerate until ready to serve.
Preheat a grill or grill pan to medium-high heat (about 400°F). Remove chicken from marinade and grill 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 rice among four shallow bowls. Top with sliced chicken, cherry tomatoes, cucumber, red onion, and crumbled feta. Drizzle with yogurt sauce and garnish with fresh dill and lemon wedges.