  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 $\mathfrak{h}(M)$ becomes a direct summand of a motive $\bigoplus
\mathfrak{h}(S^{k_i})(n_i)$ with $k_i\leq \dim(M)$. The result implies that
finite dimensionality of $\mathfrak{h}(M)$ follows from finite dimensionality
of $\mathfrak{h}(S)$. The technique also applies to moduli spaces of twisted
sheaves and to moduli spaces of stable objects in ${\rm D}^{\rm b}(S,\alpha)$
for a Brauer class $\alpha\in{\rm Br}(S)$. 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 $\widetilde H(S,\alpha,\mathbb{Z}) \simeq
\widetilde H({\cal A}_X,\mathbb{Z})$. In this case, we prove that there is an
isomorphism of transcendental Chow motives $\mathfrak{t}(S)(1) \simeq
\mathfrak{t}(X)$.

Algebraic Geometry