It is also can be seen from the independence=>zero correlation result, that independence can be equivalently written as X and Y are independent if E[g(X)f(Y)] = E[g(X)]E[f(Y)] for each pair of bounded and continuous functions g,f. It is clear then why zero correlation, or E[XY]=E[X]E[Y] is not sufficient for independence.
A uniformly-distributed point on a chessboard is a nice example for this and some related things. - colour, row, and column are pairwise independent, but row and column completely determine colour. - colour is uncorrelated with, but perfectly determined by, diagonal position (ie, row+column)
Thanks - nice example!