In a small mixing bowl, whisk together rum, sugar, and cinnamon until completely incorporated.
Marinate pineapple and banana slices for 20 minutes in rum mixture.
Heat grill to about 350 degrees and oil grate. Place pineapple on grill, cook for 1 minute, then rotate rings about 45 degrees to produce an attractive crisscross grill pattern; continue to cook for 1 minute.
Flip pineapple rings over, glaze with rum marinade, and cook for 1 minute; rotate as described above and cook for 1 minute longer. Remove from grill.
Re-oil grate. Cut about 1 ½ inches off each end of bananas; discard ends. Slice remaining center pieces in half lengthwise.
Place slices, cut side down, on grate so grill marks will be on the diagonal. Cook for 45 seconds, rotate slices about 45 degrees, and cook for another 45 seconds.
Flip bananas over, glaze with marinade, and cook for 45 seconds; rotate and cook for another 45 seconds. Remove from grill.