In a small bowl, mix together the spice rub ingredients: smoked paprika, ground ginger, onion powder, garlic powder, cumin, salt, and pepper. Pat chicken dry with paper towels, then season the chicken pieces evenly with the spice rub mixture. Cover or place in resealable bag and refrigerate for 30-60 minutes.
Meanwhile, add the sauce ingredients to a medium saucepan. Simmer, stirring occasionally, until sauce has thickened slightly, about 10-15 minutes. Remove and reserve 1/2 cup of sauce for serving. Use remaining sauce to baste chicken while grilling.
Turn all gas grill burners to high, close lid, and heat for about 15 minutes. Scrape the grill grates clean. Leave primary burner(s) on, turn off other burners. Oil the grill grates. (I use tongs, a paper towel, and cooking oil.)
Thread the chicken pieces onto skewers. Grill over the primary burner(s) with lid closed, turning one-quarter of a turn every 2 minutes or so, until browned and slightly charred (about 8 minutes total for chicken breast and 10 minutes for thighs). Brush the top of kebabs with Huli Huli sauce during cooking.
At the end of cooking time, brush top of kebabs with more of the sauce, flip and cook about 1 minute. Brush second side with sauce; flip and cook for another minute or so.
Remove kebabs from grill, tent with foil, and let rest for 5 minutes. Serve with reserved Huli Huli sauce.