  We show that any nonconstant morphism of a threefold admits a relative
Chow-Kuenneth decomposition. As a corollary we get sufficient conditions for
threefolds to admit an absolute Chow-Kuenneth decomposition. In case the image
of the morphism is a surface, this implies another proof of a theorem on the
absolute Chow-Kuenneth decomposition for threefolds satisfying a certain
condition, which was obtained by the first author with P. L. del Angel. In case
the image is a curve, this improves in the threefold case a theorem obtained by
the second author where the singularity of the morphism was assumed isolated
and the condition on the general fiber was stronger.
