Based on the recent mathematical theory of isotopic tilings, we present the, to the best of our knowledge, first algorithm for the enumeration of isotopy classes of cellular embeddings of graphs invariant under a given symmetry group on hyperbolic surfaces. To achieve this, we substitute the isotopy classes with combinatorial objects and propose different techniques, guided by structural results on the mapping class group of an orbifold and notions from computational group theory that ensure that the algorithm is computationally tractable. Furthermore, we extend data structures of combinatorial tiling theory to isotopy classes that lead to an actual implementation of the algorithm for symmetry groups generated by rotations. \\ From the enumerated combinatorial objects, we produce a range of simple graphs on hyperbolic surfaces represented as symmetric tilings in the hyperbolic plane, illustrating the enumeration with examples and experimentally demonstrating the feasibility of the approach. These tilings are finally projected onto a family of triply-periodic surfaces that are relevant for the natural sciences.