Rub chicken breasts with olive oil, dried oregano, salt, and pepper.
Grill or pan-sear chicken over medium-high heat for approximately 6 to 7 minutes per side until the internal temperature reaches 165°F (74°C).
Remove chicken from heat and allow to rest for 10 minutes before dicing into 1/2-inch cubes.
Grate one half of the English cucumber; place in a clean kitchen towel and squeeze firmly to remove all excess moisture.
Dice the remaining half of the cucumber into 1/2-inch pieces and set aside.
In a large mixing bowl, whisk together the Greek yogurt, squeezed grated cucumber, minced garlic, lemon juice, and fresh dill to create the tzatziki base.
Incorporate the diced chicken, diced cucumber, red onion, olives, and tomatoes into the yogurt mixture.
Gently fold in the crumbled feta cheese to maintain its structure.
Refrigerate for at least 30 minutes to allow flavors to meld before serving.