Whisk 1/2 cup of the yogurt with the garlic, 1 tablespoon lemon juice, half the lemon zest, the olive oil, oregano, salt, and pepper in a large bowl.

Add the chicken cubes and toss until every piece is coated, then cover and refrigerate for at least 4 hours, or up to overnight.

If using wooden skewers, soak them in water for 20 minutes before threading so they do not scorch.

Thread 4 to 5 pieces of chicken onto each skewer, leaving a little space between pieces so the heat reaches all sides.

Heat a grill or grill pan to medium-high and cook the skewers for 5 to 6 minutes per side, until the chicken is opaque throughout and a thermometer in the thickest piece reads 165 degrees F.

Transfer the cooked skewers to a plate and let them cool to room temperature, about 20 minutes.

Stir the remaining yogurt with the remaining lemon juice and zest and the dill to make the dipping sauce.

Cover the skewers and the sauce separately and refrigerate until serving; they are meant to be eaten cold or at room temperature, not reheated.