Leave root and 1 inch stem on beets; scrub with a brush. Place in a medium saucepan; cover with water. Bring to a boil; cover, reduce heat, and simmer 35 minutes or until tender. Drain and rinse with cold water, and cool. Trim off beet roots, and rub off skins. Cut beets into 1/4-inch-wide wedges.
Bring the chicken broth to a boil in a large saucepan. Add the barley; cover, reduce heat, and simmer 40 minutes. Remove from heat; cool.
Combine the barley, arugula, walnuts, and cheese in a large bowl. Combine vinegar and remaining ingredients in a small bowl. Pour the vinegar mixture over the barley mixture, tossing to coat. Top with beets.