Combine all ingredients to make the marinade. Pour marinade over the turkey breast in a non-reactive bowl (glass, ceramic or stainless steel) and refrigerate for at least 1 hour up to overnight.
When ready to grill, remove the turkey breast from the marinade and wipe off any excess. Discard any remaining marinade.
Combine all spices and herbs to make a rub. Rub generously all over the turkey breast. Season with salt and pepper to taste.
Grill the turkey breast over medium heat, with the lid closed, 15-20 minutes per side, until cooked through and no longer pink and a digital thermometer inserted into the thickest part of the meat registers 165°F. Let the meat rest for at least 15 minutes before slicing.
For the salsa, combine all ingredients in a mini blender or chopper. Pulse until roughly chopped. Season with salt and pepper to taste.
Serve sliced turkey immediately with salsa on the side.