Zest and juice half the lemon into a bowl. Add 2 tbsp olive oil, 1 tsp dried oregano, and a generous pinch of salt and pepper. Toss the chicken in the marinade and set it aside for 10 minutes.
In a small pot, combine 1 cup rice with 1.5 cups water and a pinch of salt. Bring to a boil, cover, reduce to low, and cook for 15 minutes. Turn off the heat and let it sit, covered, for 5 more minutes. Fluff with a fork.
Grate half the cucumber using the large holes of a box grater. Toss with 1/2 tsp salt in a small bowl. Let sit 5 minutes, then squeeze out the liquid using a clean kitchen towel. Mix the squeezed cucumber with 1 cup Greek yogurt, 1 grated garlic clove, 1 tbsp lemon juice, and a pinch of chopped fresh mint. Season to taste.
Heat 1 tbsp olive oil in a cast-iron skillet over medium-high heat until shimmering. Add the chicken in a single layer (work in batches if needed). Cook without moving for 5-6 minutes until deeply golden, then flip and cook another 5-6 minutes until internal temp reaches 165°F.
Transfer the chicken to a cutting board and let rest for 5 minutes. Meanwhile, toss the remaining diced cucumber, halved cherry tomatoes, crumbled feta, and olives with a drizzle of olive oil, a splash of lemon juice, and a pinch of oregano.
Slice the chicken against the grain into thick strips. Divide the rice among four bowls, top with the chicken, pile on the salad, and finish with a generous dollop of tzatziki. Serve immediately.