Kalinic did not make a great impact at Ewood, partly because of the style of play, partly because of injury and partly through being left out of the side.
The Croatia international, who arrived at the Premier League club from Hadjuk Split two years ago, leaves Ewood Park for an undisclosed fee.
He played for Croatia as a substitute in Wednesday night's international draw with the Republic of Ireland in Dublin and last scored for Blackburn in a 4-1 defeat at Aston Villa in February.