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Combine the butter, brown sugar, rum, cinnamon, and salt in a saucepan and bring to a boil over high heat. Let the glaze boil until thick and syrupy, about 5 minutes.
Prepare and preheat the grill to high. Brush and oil the grate. Next, place the skewered peaches on the hot grate and grill until nicely browned, 3 to 4 minutes per side, basting with the rum and butter glaze. Spoon any remaining glaze over the grilled peaches and serve at once. Peach or vanilla ice cream makes a great accompaniment.