Sift flour, baking powder, and salt together, set aside. In a large bowl, cream together the margarine and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla. Gradually blend in the sifted ingredients until fully absorbed. Cover dough, and chill for 2 hours.
Preheat oven to 400 degrees F (200 degrees C). Grease cookie sheets. On a clean floured surface, roll out small portions of chilled dough to 1/4 inch thickness. Cut out shapes using cookie cutters.
Bake 6 to 8 minutes in the preheated oven, or until edges are barely brown. Remove from cookie sheets to cool on wire racks.