  This is the first article in a series of two papers in which we study the
Temperleyan dimer model on an arbitrary bounded Riemann surface of finite
topolgical type. The end goal of both papers is to prove the convergence of
height fluctuations to a universal and conformally invariant scaling limit. In
this part we show that the dimer model on the Temperleyan superposition of a
graph embedded on the surface and its dual is well posed, provided that we
remove an appropriate number of punctures. We further show that the resulting
dimer configuration is in bijection with an object which we call Temperleyan
forest, whose law is characterised in terms of a certain topological condition.
Finally we discuss the relation between height differences and Temperleyan
forest, and give a criterion guaranteeing the convergence of the height
fluctuations in terms of the Temperleyan forest.
