Step 1: In a bowl, whisk together olive oil, lemon juice, minced garlic, oregano, salt, and pepper to create the marinade.
Step 2: Add chicken cubes to the marinade, toss to coat, cover, and refrigerate for at least 30 minutes (up to 4 hours).
Step 3: While chicken marinates, prepare tzatziki by combining grated cucumber, Greek yogurt, dill, lemon juice, minced garlic, and salt in a bowl. Refrigerate until ready to serve.
Step 4: If using wooden skewers, soak them in water for 30 minutes to prevent burning.
Step 5: Thread marinated chicken cubes onto skewers, leaving a small space between pieces.
Step 6: Preheat grill or grill pan to medium-high heat and lightly oil the grates.
Step 7: Grill skewers for 10-12 minutes, turning occasionally, until chicken is cooked through and reaches 165°F internally.
Step 8: Serve hot skewers with chilled tzatziki sauce on the side.