Juice fresh oranges and limes using a citrus juicer, then combine the juices in a measuring cup.
Pour the citrus juice into the blender, followed by the rum and simple syrup. Add the ground cinnamon for a hint of spice.
Add the ice cubes to the blender to create a chilled, slushy mixture.
Blend on high speed for 30-45 seconds until the mixture is smooth and icy, with a texture similar to granular snow.
Stop the blender and check the texture; it should be thick and slushy but pourable. If needed, blend a few seconds longer.
Pour the frosty mixture into glasses, garnishing with a citrus slice or a sprinkle of cinnamon if desired.
Enjoy immediately with a straw or spoon for a cool, refreshing treat that captures summery zest with a spirited kick.