Combine the ginger with the sugar and 1/2 cup water in a small saucepan, bring to a boil, and then reduce the heat and simmer for 10 minutes. Allow the mixture to cool, then strain, reserving the sliced ginger.
Put 2 tablespoons each of the ginger syrup in 4 tall glasses, add a mint sprig to each and a couple slices of ginger; slightly muddle or crush. Fill the glasses 3/4 of the way with ice cubes. Add 1/2 cup of orange juice and 1 tablespoon of lime juice to each glass and stir. Top off each glass with 1/2 cup seltzer and stir well. Serve.