Place the grapes in a 13 x 9-inch baking pan. Freeze 2 hours.
Transfer grapes to a food processor. Add pomegranate juice, lime juice and sugar. Pulse until fairly smooth. Return mixture to baking pan in even layer and place back in freezer. Freeze 1 hour.
Stir with a fork, bringing the firmer mixture at the edges toward the center. Freeze again for 30 minutes.
When it is firm enough to serve, scoop granita into glasses. Serve cold with a dollop of yogurt and some mint, champagne grapes or small red grapes or pomegranate seeds, as desired.