Bring currants and rum to a boil in a saucepan. Remove from heat; cover and let stand 30 minutes. Drain currants, discarding rum.
Beat butter at medium speed with an electric mixer until creamy; gradually add sugar, beating well.
Combine flour, baking powder, and salt; gradually add to butter mixture, beating at low speed until blended after each addition. Stir in currants. Chill 30 minutes.
Roll dough to 1/4-inch thickness on a floured surface. Cut with a 2-inch round cutter; place 2 inches apart on greased baking sheets.
Bake at 375° for 12 minutes or until edges just begin to brown; cool on baking sheets on wire racks 5 minutes. Remove to wire racks to cool completely.