The surprise ingredient here is balsamic vinegar.
Set a colander over a bowl, and drain cherries, reserving all the liquid.
Place cherries in a food processor or in a blender, and puree. Pass the puree through a fine sieve, set over a medium bowl. Using the back of a spoon, press cherries to extract the juice. If any solids remain in juice, pass juice through the sieve again. Discard pulp left in the sieve.
Add reserved liquid from step one and vinegar to the strained juice.
In a medium saucepan, combine sugar with 2/3 cup water; cook over medium-high heat until sugar has dissolved, about 4 minutes. Remove the pan from heat, and let stand until cool.
Add cooled syrup to juice mixture, and stir until combined. Pour this new mixture into a shallow metal pan large enough to hold the liquid; place in freezer until ice crystals begin to form around edges, about 1 hour. Using a fork, stir the ice crystals into the center; return to the freezer. Stir ice crystals once an hour until all the liquid has frozen and has a grainy consistency, about 5 hours. Serve, or store in the freezer, covered, up to 2 days.
Stirring the granita frequently as it freezes gives it a granular texture.