On the Griffiths group of the cubic sevenfold

We prove that the Griffiths group of 3-cycles homologous to zero modulo algebraic equivalence, on a generic hypersurfaces of dimension 7 and degree 3 is not finitely generated, even when tensored with Q. Using this and a result of Nori, we give examples of varieties for which some Griffiths group is not finitely generated (modulo torsion) but whose corresponding intermediate Jacobian is trivial.