In a large bowl, whisk oil, lime juice, soy sauce, brown sugar, and cumin until combined. Stir in cilantro; season with salt and pepper. Set aside half of dressing for serving.
Add chicken, ginger, and garlic to remaining dressing and toss to coat. Cover and let marinate at least 30 minutes at room temperature or up to 4 hours in the refrigerator.
Meanwhile, if using wooden skewers, soak skewers in water at least 30 minutes before grilling.
Prepare a grill for medium-high heat; preheat 5 minutes (or heat a grill pan over medium-high heat). Carefully thread chicken onto skewers; season with salt and pepper. Discard remaining marinade.
Grill chicken, turning occasionally, until golden brown and an instant-read thermometer inserted into thickest part registers 165°, about 5 minutes per side.
Transfer skewers to a plate. Spoon some reserved dressing over top. Serve with remaining dressing and lime wedges alongside.
