In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to obtain integrable dimer models. These are dimer models on doubly periodic bipartite graphs with quasi-periodic positive weights. Dimer models with periodic weights and Harnack curves are recovered as a special case. This generalization from Harnack curves to general M-curves leads to transparent algebro-geometric structures. In particular explicit formulas for the Ronkin function and surface tension as integrals of meromorphic differentials on M-curves are obtained. Furthermore we describe the variational principle for the height function in the quasi-periodic case. Based on Schottky uniformizations of Riemann surfaces we present concrete computational results including computing the weights and sampling dimer configurations with them. The computational results are in complete agreement with the theoretical predictions.