For any smooth projective moduli space M of Gieseker stable sheaves on a complex projective K3 surface (or an abelian surface) S, we prove that the Chow motive hpM q becomes a direct summand of a motive À hpS k i qpniq with ki ď dimpM q. The result implies that finite dimensionality of hpM q follows from finite dimensionality of hpSq. The technique also applies to moduli spaces of twisted sheaves and to moduli spaces of stable objects in D b pS, αq for a Brauer class α P BrpSq. In a similar vein, we investigate the relation between the Chow motives of a K3 surface S and a cubic fourfold X when there exists an isometry r HpS, α, Zq » r HpAX , Zq. In this case, we prove that there is an isomorphism of transcendental Chow motives tpSqp1q » tpXq.

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