My mother used to make this as watermelons were plentiful and inexpensive. This recipe is very close to what I remember it tasting like.
8 cups watermelon, cut in 2 inch pieces, seeded
Puree watermelon in small batches in the blender then strain into a pitcher pressing on the solids to extract as much liquid as possible. Now you have a choice, some people like the solids added to the drink and others don't. Do what you like. Pour the watermelon into a pitcher, add simple syrup, lemon and lime juice and grenadine and stir to blend. Put in refrigerator to chill for at least 1 hour.