Preheat a grill to medium-high heat or preheat the oven to 375°F (190°C).
Make the marinade: in a food processor or blender, combine garlic, roma tomatoes, juice of 1 lemon, 1/2 tsp salt, 1/2 tsp pepper, oregano, cinnamon, allspice, 1/4 cup parsley/cilantro, 2 tbsp olive oil, and 1/4 cup yogurt. Pulse or blend until smooth.
Toss the 1-inch chicken cubes with the prepared marinade in a bowl until evenly coated. Let sit at room temperature 10 minutes or refrigerate up to a few hours.
Thread marinated chicken onto soaked wooden skewers, placing a thin lemon slice between some cubes as desired.
Grill the skewers 3–5 minutes per side with the lid closed, until chicken is cooked through and juices run clear (internal temperature 165°F / 74°C). If using the oven, bake at 375°F for about 12–18 minutes, turning once, until cooked through.
Transfer cooked skewers to a platter and sprinkle generously with sumac.
Make the yogurt drink: combine 2 cups yogurt, 1/2 cup cold water, 1/2 cup ice, 1/4 tsp salt, and 1/2 cup mint and parsley in a blender. Blend until smooth and slightly foamy, then chill immediately.
Serve the skewers hot with the chilled yogurt drink on the side.