Whirl rice in a food processor to break into small pieces. Pour into a bowl and add 3 1/2 cups water, cinnamon stick and cardamom pods. Cover and let stand overnight.
Stir in sugar and vanilla. Purée half in batches in a blender until very smooth, about 1 1/2 minutes. Strain twice, through 3 layers of damp cheesecloth and then through a very fine strainer, into a pitcher. Pour over ice.
Note: Nutritional analysis is per 3/4-cup serving.