Turn on a gas grill to medium-high or light a charcoal grill and let the coals burn until quite hot and covered with white ash. Spread the mussels onto the grill grates. As they open - it'll take 2 to 4 minutes - they're ready when the two sides of the shell have released completely from each other - remove the mussels to a rimmed baking sheet. When cool enough to handle, break off the empty half of the shell and, using your fingers or a small knife, pull the meat from the shell. Lay the meat back in the shell (the empty side of the shell is the prettiest for serving).
Arrange the mussels on a serving platter. Spoon a half teaspoon or so of the salsa on each one, sprinkle with cilantro and you're ready to serve.