Does Cauchy's integral theorem work for Riemann Surfaces?

Question

The way I learned to prove Cauchy's Integral Theorem was by using Green's Theorem and Cauchy-Riemann equations. Now that we're working with Riemann-Surfaces, doing a simple closed curve might not necessary get you back at the same point on the Surface. I've experimented with some functions like $\sqrt{z}$ and $\sqrt[3]{z}$ and figured out if I looped around 2 and 3 times respectively, the integral would be 0. How can I prove it? I'm trying to apply green's theorem, but the nature of the Riemann-Surface makes it hard to comprehend the double integral.