Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L\subset X to the skyscraper sheaf of a point y\in Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothedieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville-Voisin' subring in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.