Granita, or Italian Ice, is a simple and delicious dessert made with only 4 ingredients. One of them is often the traditional, beloved Cherokee fruit, strawberries. June is strawberry season. Use locally grown strawberries or pick-yourself strawberries for the best flavor. Large supermarket strawberries have only a hint of strawberry flavor and often have a grainy texture because their natural sugar transforms to starch as soon as they are picked.