We develop a framework to study the dimer model on Temperleyan graphs embedded on a Riemann surface with finitely many holes and handles. We extend Temperley's bijection to this setting and show that the dimer model can be understood in terms of an object which we call Temperleyan forests. Extending our earlier work to the setup of Riemann surfaces, we show that if the Temperleyan forest has a scaling limit then the fluctuations of the height one-form of the dimer model also converge. Furthermore, if the Riemann surface is either a torus or an annulus, we show that Temperleyan forests reduce to cycle-rooted spanning forests and show convergence of the latter to a conformally invariant, universal scaling limit. This generalises a result of Kassel--Kenyon. As a consequence, the dimer height one-form fluctuations also converge on these surfaces, and the limit is conformally invariant. Combining our results with those of Dub\'edat, this implies that the height one-form on the torus converges to the compactified Gaussian free field, thereby settling a question in \cite{DubedatGheissari}. This is the first part in a series of works on the scaling limit of the dimer model on general Riemann surfaces.