In a large bowl, whisk together 2 tablespoons olive oil, 1 tablespoon lemon juice, oregano, half the garlic, salt, and pepper. Add chicken and toss to coat; marinate for 15 minutes.
In a medium saucepan, combine quinoa and chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 15 minutes or until liquid is absorbed. Remove from heat and let stand covered for 5 minutes.
Fluff the quinoa with a fork and stir in 1 tablespoon of lemon juice and remaining olive oil.
Heat a large skillet over medium-high heat. Add the marinated chicken and cook for 8 to 10 minutes, stirring occasionally, until browned and the internal temperature reaches 165°F (74°C).
In a separate bowl, toss the cucumber, tomatoes, red onion, and olives together.
Prepare the sauce by mixing the Greek yogurt, dill, remaining lemon juice, and remaining garlic in a small ramekin.
Assemble the bowls by dividing the lemony quinoa equally among four containers. Top with the cooked chicken, vegetable mixture, and crumbled feta cheese. Serve with the yogurt sauce on the side.