So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper (http://arxiv.org/pdf/1007.5467.pdf). Basically, you end up getting:
$$ \begin{equation} K_M(\textbf{x}, \textbf{y}, t) = \sum\limits_{g \in G} K_\tilde{M}( \tilde{\textbf{x}}, g \cdot \tilde{\textbf{y}}, t) \end{equation} $$
where $\tilde{M}$ is the universal cover of $M$ and $G$ is the covering group. I'm wondering if you can do a similar thing with a fundamental solution of the Laplace equation, which I'd really like to calculate for arbitrary Riemann surfaces (actually, compact hyperbolic surfaces).