  We prove several results concerning the existence of surfaces of section for
the geodesic flows of closed orientable Riemannian surfaces. The surfaces of
section $\Sigma$ that we construct are either Birkhoff sections, meaning that
they intersect every sufficiently long orbit segment of the geodesic flow, or
at least they have some hyperbolic components in $\partial\Sigma$ as limit sets
of the orbits of the geodesic flow that do not return to $\Sigma$. In order to
prove these theorems, we provide a study of configurations of simple closed
geodesics of closed orientable Riemannian surfaces, which may have independent
interest. Our arguments are based on Grayson's curve shortening flow.
