  In this paper we are concerned with the Riesz transform on the direct product
manifold ${\mathbb{H}}^n \times M$, where ${\mathbb{H}}^n$ is the
$n$-dimensional real hyperbolic space and $M$ is a connected complete
non-compact Riemannian manifold satisfying the volume doubling property and
generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We
establish its weak type $(1,1)$ property. In addition, we obtain the weak type
$(1, 1)$ of the heat maximal operator in the same setting. Our arguments also
work for a large class of direct product manifolds with exponential volume
growth. Particularly, we provide a simpler proof of weak type $(1,1)$
boundedness of some operators considered in the work of Li, Sj\"ogren and Wu
[27].
