We consider spacelike graphs $\Gamma_f$
of simple products $(M\times N, g\times -h)$
where $(M,g)$ and $(N,h)$ are Riemannian
manifolds and $f:M\rightarrow N$ is a smooth map.
Under the condition of the Cheeger constant of $M$ to be zero and some
condition on the second fundamental form at infinity,
we conclude that
if $\Gamma_f\subset M\times N$ has parallel mean curvature $H$
then $H=0$.
This holds trivially if $M$ is closed.
If $M$ is the $m$-hyperbolic space then
for any constant $c$, we describe an explicit foliation
of $ {\mathbb H}^m\times \mathbb R$ by hypersurfaces with constant mean curvature
$c$.