In a large pitcher, layer the pineapple, strawberries, orange slices and mint sprigs. Pour in the pineapple juice, orange juice and Triple Sec. With a wooden spoon, carefully mix together and slightly muddle the fruit. Place in the refrigerator for at least 1 hour.
When ready to serve, spoon 1/4 of soaked fruit into each glass, then fill glass up 1/3 full with juice. Top with Prosecco and garnish with a mint sprig.