Cut up a lemon, and squeeze the juice over the stain. Rub the inside of the lemon all over the stains, rinse the bowl out and take it outside. Leave it sitting in the sun for a few hours, and then rinse it again. The sun is the most natural way to bleach things.
Soak the bowl in some bleach, and scrub the stain with a sponge. Rinse the bleach away with cold water. Run the bowl through the dishwasher.
Squirt some dishwashing liquid onto the stain, and add enough baking soda to make a paste. Scrub the stain with the cleaning mixture. The baking soda will add some heft to the dishwashing liquid and help power the stains out.
Scrub the stains out using a sponge dampened with white vinegar. Soak the bowl in a sink filled with hot water and 1/2 cup vinegar for half an hour, if necessary. Rinse the bowl clean with cold water. Vinegar is a natural way to remove stains.