Preheat a grill. In a bowl, blend rum, honey, and cinnamon. Reserve.
Over direct medium heat, grill bananas, cut side down for 3 minutes, or until the bananas develop visible grill marks.
Turn bananas and brush with honey-rum mixture. Continue grilling, with lid down, for 5 minutes, or until bananas are cooked throughout.
Remove, discard peel, and serve immediately with ice cream.