We present a technique for the enumeration of all isotopically distinct ways of tiling, with disks, a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group of the tiling under consideration as a set of algebraic operations on certain generators of the symmetry group. We derive explicit descriptions of certain subgroups of mapping class groups and of tilings as embedded graphs on orbifolds. We further use this explicit description to present an algorithm that we illustrate by producing an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the prominent Primitive, Diamond and Gyroid triply-periodic minimal surfaces, outlining how the approach yields an unambiguous enumeration. We also present the corresponding 3-periodic graphs on these surfaces.