Of the several existing algorithms for computing Delaunay triangulations of point sets in Euclidean space, the incremental algorithm has recently been extended to the Bolza surface, a hyperbolic surface of genus 2. We will generalize this algorithm to so called symmetric hyperbolic surfaces of arbitrary genus. Delaunay triangulations of point sets on hyperbolic surfaces can be constructed by using the fact that such point sets can be regarded as periodic point sets in the hyperbolic plane. However, one of the main issues is then that the result might contain 1-or 2-cycles, which means that the triangulation is not simplicial. As the incremental algorithm that we use can only work with simplicial complexes, this situation must be avoided. In this work, we will first compute the systole of the symmetric hyperbolic surfaces, i.e., the length of the shortest non-contractible loop. The value of the systole is used in a condition to ensure that the triangulations will be simplicial. Secondly, we will show that it is sufficient to consider only a finite subset of the infinite periodic point set in the hyperbolic plane. Finally, we will algorithmically construct a point set with which we can initialize the algorithm.