  Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map
induces a morphism from each smooth variety parameterizing 1-cycles in Y to the
intermediate Jacobian J(Y). We study in this paper the existence of families of
1-cycles in Y for which this induced morphism is surjective with rationally
connected general fiber, and various applications of this property. When Y
itself is rationally connected, we relate this property to the existence of an
integral homological decomposition of the diagonal. We also study this property
for cubic threefolds, completing the work of Iliev-Markoushevich. We then
conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on
fibrations into cubic threefolds over curves, with restriction on singular
fibers.
