In a small bowl, whisk together the olive oil, half of the lemon juice, oregano, garlic powder, sea salt, and black pepper.
Slice the chicken breast into thin strips and toss in the marinade; let sit for 10 minutes.
Preheat a grill pan or outdoor grill over medium-high heat and cook the chicken for 4-5 minutes per side until charred and cooked through.
While the chicken cooks, grate a portion of the cucumber and squeeze out excess moisture using a paper towel.
In a small bowl, combine the Greek yogurt, grated cucumber, remaining lemon juice, and chopped fresh dill to make the tzatziki.
Dice the remaining cucumber, halve the cherry tomatoes, and thinly slice the red onion.
Warm the pita slightly on the grill for 30 seconds per side.
Assemble the gyro by spreading the tzatziki on the pita, adding the grilled chicken, and topping with the fresh vegetables.