  Let S be a K3 surface obtained as triple cover of a quadric branched along a
genus 4 curve. Using the relation with cubic fourfolds, we show that S has
finite dimensional motive, in the sense of Kimura. We also establish the
Kuga-Satake Hodge conjecture for S, as well as Voisin'conjecture concerning
zero-cycles. As a consequence, we obtain Kimura finite dimensionality, the
Kuga-Sataka Hodge conjecture, and Voisin's conjecture for 2 (9-dimensional)
irreducible components of the moduli space of K3 surfaces with an order 3
non-symplectic automorphism.
