Prepare pineapple by cutting off the outside skin, being careful not to leave any green or brown husk. Cut the pineapple into rings.
Cut the peaches in half and remove the pit.
In a large bowl, toss pineapple and peaches with oil, cinnamon and sugar. Place each piece on the grill and grill until caramelization appears on the fruit and the internal temperature reaches 140F.
After grilling, cut peach halves into slices. Serve about 1 pineapple slice and 3-4 peach slices per portion.