I do a few different kinds of chicken breasts and chicken thighs (boneless) on the grill. If you like East Asian cooking, yakitori is a delicious way to eat grilled chicken. Traditionally, yakitori uses skewered pieces of chicken, but you can use the basting sauce I'm about to tell you about on whole chicken breasts. Make a sauce containing soy sauce, sake, mirin, and some kind of sugar. If you can't find a good, authentic mirin (don't get the stuff made out of corn syrup), you can add more sugar and more sake. All you do for your resulting "tare" (tah-ray) is bring it to a boil then simmer until the sauce thickens. The sugar's there to give it a viscous consistency. I usually do a ratio of 2 parts soy sauce, 1 part sake, and 1 part mirin. For the sugar, I usually do 1/4 cup of sugar per 8 fl oz of soy sauce but you can adjust the sugar to taste. Honey is a good substitute for sugar, but I don't have a good quantity of honey handy. The sugar's supposed to thicken your sauce, so you'll have a very thin sauce if you omit it altogether.
When getting your chicken breasts on the grill, sear both sides before cooking it at a lower heat. Once you turn down the heat, this is where you start basting your chicken. Because the sugars in your tare will be subject to the maillard reaction when you're basting your chicken, you want a lower temperature to reduce burning while basting. Depending on how quickly your grill can go down in temperature, you may want to remove the breasts briefly before the temperature is low enough to where your tare won't burn from flare-ups. When your chicken has an internal temperature of 165F (76C), your chicken is ready to eat. Your mileage may vary, but my chicken normally finishes around 15-20 minutes. Chicken is better overcooked than undercooked. Getting it just right is a difficult skill depending on your equipment and your conviction, but you don't wanna get sick from taking it off too soon.
The tare sauce recipe I described is meant to be a base sauce. You can further spruce your tare up with (non-)traditional Japanese flavorings like shichimi togarashi, dashi stock, ginger paste, wasabi, garlic, black pepper, onion powder, green onions, sesame seeds, the sky's the limit. If you're trying to watch your sodium intake, buy a reduced sodium soy sauce for your recipe. There are also gluten-free soy sauces in case you're cooking for anyone with a gluten sensitivity.