Pat the chicken dry and rub with the olive oil, dried oregano, garlic powder, salt, and black pepper. Let marinate while preparing the remaining ingredients.
Make the dill yogurt sauce by combining the Greek yogurt, fresh dill, garlic, half the lemon juice, and half of the dried oregano. Season with black pepper, then set aside.
Place the chickpeas in a bowl and press with a fork until about half are broken open. Toss the smashed chickpeas with remaining dried oregano and lemon juice. Smashing the chickpeas helps keep them from rolling out of the wrap.
Heat a grill pan or skillet over medium heat. Cook the chicken on both sides to an internal temperature of 165°F (74°C). Let rest for 5 minutes, then slice into thin strips.
Warm the tortilla wraps in a dry skillet for about 20 seconds per side until soft and pliable.
Spread the hummus evenly over each tortilla. Layer with romaine lettuce, seasoned smashed chickpeas, sliced chicken, cucumber, tomato, and red onion. Drizzle with the dill yogurt sauce. Fold in the sides and roll the wraps tightly. Slice each wrap in half.
Stir together the sparkling water, lemon, and chia seeds. Let stand for 10 minutes, then stir again. Add ice before serving.
Serve the Greek chickpea wraps with potato chips, pickle spears, and lemon chia sparkling water.