Place chicken in a plastic bag with oil, rosemary and garlic, season well, seal, then massage chicken through bag to coat well. Refrigerate to marinate (at least 2 hours or overnight). Bring to room temperature an hour before cooking.
Heat a barbecue to medium-high heat and grill chicken, turning often, until golden and the juices run clear when the thigh is pierced with a skewer (20-25 minutes).
For salmoriglio, pound garlic and a pinch of salt to a paste with a mortar and pestle. Add oregano, pound to a paste, then stir in olive oil and lemon juice and season to taste. Serve with chicken and lemon halves on the side.