Combine oil, lemon juice, garlic, and oregano in large bowl. Slice eggplant into 1/2-inch-thick rounds. Cut small squash lengthwise into halves; cut large squash into 1/2-inch thick pieces. Cut bell peppers into large chunks. Cut onions into wedges or thick slices.
Toss vegetables with oil mixture to coat. Place vegetables in single layer on grid; reserve remaining oil mixture. Grill on covered grill 10 to 20 minutes or until tender, turning once and basting with remaining oil mixture.