A compact Riemann surface with a regular dessin on it is called a quasiplatonic curve. It is not known how many such curves there are for arbitrary genus greater than one, although there are only finitely many regular dessins for each genus. Based on theorems of Jones and Wolfart, we will discuss how to count regular dessins with a cyclic group of automorphisms which also acts on the quasiplatonic curve it is embedded on. That is, we count the number of torsion-free Fuchsian groups of index p which are normally contained in a hyperbolic triangle group of signature (p,p,p) for a prime p greater than three. We will also describe the role of the combinatorics of the cyclic regular dessins.