In a bowl, whisk together olive oil, lemon juice, minced garlic, dried oregano, cumin, salt, and pepper. This will be your marinade that infuses the chicken with bright, savory notes.
Dice or slice the chicken thighs into evenly sized pieces, about 1 to 1.5 inches. Toss them in the marinade until all pieces are well coated. Cover and refrigerate for at least 20 minutes, up to 2 hours, to let the flavors sink in.
While the chicken marinates, make the tzatziki. In a bowl, combine Greek yogurt, grated cucumber (squeezed dry to remove excess moisture), chopped dill, minced garlic, lemon juice, and a pinch of salt. Stir until smooth, then cover and refrigerate for at least 15 minutes to let the flavors meld.
If using wooden skewers, soak them in water for 15 minutes to prevent burning on the grill. Thread marinated chicken pieces onto skewers, leaving a little space between each piece for even grilling.
Preheat your grill to medium-high heat. Lightly oil the grates to prevent sticking. Grill the skewers, turning every 2–3 minutes, until the chicken is cooked through and has a nice char, about 8–12 minutes depending on thickness.
Let the skewers rest for a couple of minutes after grilling to retain juiciness. Serve warm with a generous dollop of homemade tzatziki on the side or spooned over the top.
Accompaniments can include warm pita, sliced fresh tomatoes, a cucumber salad, or a lemony herbed rice to complete the meal.