Mix the marinade: In a bowl, whisk together the olive oil, 2 tablespoons lemon juice, minced garlic, oregano, 1/2 teaspoon salt, and black pepper. Add the sliced chicken and toss to coat evenly.
Let chicken rest: Let the chicken sit in the marinade for 5 to 10 minutes while you prep the vegetables. This short rest is enough time for the lemon and garlic to flavor the meat.
Heat the skillet: Warm a large skillet over medium-high heat for about a minute until it's hot but not smoking. A hot pan helps the chicken sear instead of steam.
Sear the chicken: Add the chicken strips in a single layer and cook for 3 to 4 minutes per side, until golden on the outside and 165°F in the center. Avoid crowding the pan so the edges brown instead of releasing liquid.
Rest the chicken: Transfer the chicken to a plate and let it rest for 3 minutes before slicing. This keeps the juices inside the meat instead of pooling on the cutting board.
Make the tzatziki: In a bowl, stir together the Greek yogurt, half the grated cucumber, remaining 1 tablespoon lemon juice, and remaining 1/2 teaspoon salt. Taste and adjust lemon or salt as needed.
Prep the toppings: Dice the remaining cucumber, halve the cherry tomatoes, and thinly slice the red onion. Set these aside in small bowls for easy assembly.
Warm the rice: If your rice has cooled, reheat it in the microwave for 1 to 2 minutes or warm it in a saucepan over low heat, stirring occasionally.
Build the bowls: Divide the rice among four bowls, then top each with sliced chicken, cucumber, tomatoes, and red onion. Arrange the toppings in sections so the colors stay distinct.
Finish and serve: Spoon the tzatziki over the top of each bowl and scatter with crumbled feta and fresh dill if using. Serve right away while the chicken is still warm.