  In this paper, we study biharmonic Riemannian submersions $\pi:M^2\times\r\to
(N^2,h)$ from a product manifold onto a surface and obtain some local
characterizations of such biharmonic maps. Our results show that when the
target surface is flat, a proper biharmonic Riemannian submersion
$\pi:M^2\times\r\to (N^2,h)$ is locally a projection of a special twisted
product, and when the target surface is non-flat, $\pi$ is locally a special
map between two warped product spaces with a warping function that solves a
single ODE. As a by-product, we also prove that there is a unique proper
biharmonic Riemannian submersion $H^2\times \r\to \r^2$ given by the projection
of a warped product.
