Combine first 4 ingredients in a pitcher; stir well, and chill. Fill 4 glasses with orange juice mixture. Slowly pour 1 1/2 teaspoons grenadine syrup down inside of each glass (do not stir before serving).
Note: For a nonalcoholic punch, omit rum, and use 2 1/2 cups orange juice and 1 1/4 cups pineapple juice.
