A method to generate periodic geodesics in arbitrary level surfaces is presented. The underlying algorithm resolves several technical complications posed by the constraints to stay in the surface and retain periodicity. The method exploits the "inverse'' of the parallel transport equation and its "derivative.'' This approach avoids most of the complications due to the intricate form of the geodesic curvature. The process flows any periodic curve in the surface along the negative gradient trajectory of the total squared geodesic curvature. The mathematical framework is that of an infinite-dimensional Riemannian manifold representing periodic curves of arbitrary length. The method is illustrated by an example in a sphere-like surface that is neither an ellipsoid nor a surface of revolution.