Given a smooth projective
n
n
-fold
Y
Y
, with
H
3
,
0
(
Y
)
=
0
H^{3,0}(Y)=0
, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension
2
2
-cycles in
Y
Y
to the intermediate Jacobian
J
(
Y
)
J(Y)
, which is an abelian variety. Assuming
n
=
3
n=3
, we study in this paper the existence of families of
1
1
-cycles in
Y
Y
for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When
Y
Y
itself is rationally connected with trivial Brauer group, we relate this property to the existence of an integral cohomological decomposition of the diagonal of
Y
Y
. We also study this property for cubic threefolds, completing the work of Iliev-Markushevich-Tikhomirov. We then conclude that the Hodge conjecture holds for degree
4
4
integral Hodge classes on fibrations into cubic threefolds over curves, with some restriction on singular fibers.