Cook the barley with a pinch of salt according to package directions. Drain when the barley is tender to make sure all the extra water is removed. Place into a big bowl so it can cool.
While the barley's cooking, make the tomato vinaigrette. Combine all the vinaigrette ingredients in a small food processor or blender, then process until smooth.
Put 6 tablespoons of the vinaigrette into a plastic bag with the chicken (salt and pepper it first) and marinate in the refrigerator for one hour or more (24 max, and about 2-6 is the sweet spot).
For the barley corn salad, stir the corn, cherry tomatoes, and red onion with the cooled barley. Add the remaining tomato vinaigrette and stir to coat. Gently fold in the avocado, then season with salt and pepper to taste. Place into the refrigerator.
When close to eating time, heat your grill to medium-high (450 to 500 F or thereabouts) and grill the chicken for 4-5 minutes a side, or until cooked through (old recommendation was 165 F at thickest point, though more recently around 145 F minimum is recommended).
Let the chicken rest for 5 minutes off the grill, then serve with the salad---either on top of it or to the side.