Let $\Sigma_2$ be a closed genus $2$ surface. There exists an orientation-preserving diffeomorphism $f:\Sigma_2 \rightarrow \Sigma_2$ of order $3$. The diffeomorphism has $4$ fixed points (each, of course, of order $3$) and from Riemann-Hurewitz you can see that the quotient is a sphere $S^2$.
To construct $f$, it is enough to construct the appropriate branched cover of $S^2$. But this is easy: let $p_1,\ldots,p_4$ be $4$ distinct points of $S^2$ and let $X = S^2 \setminus \{p_1,\ldots,p_4\}$. There then exists a surjection $\phi:H_1(X;\mathbb{Z}) \rightarrow \mathbb{Z}/3$ which for all $1 \leq i \leq 4$ takes a loop around $p_i$ to a generator of $\mathbb{Z}/3$. Let $\widetilde{X}$ be the degree $3$ regular cover of $X$ associated to $\phi$. Then an Euler characteristic calculation shows that $\widetilde{X}$ is diffeomorphic to a genus $2$ surface minus $4$ point. The desired branched cover $\Sigma_2 \rightarrow S^2$ is then obtained by filling in these $4$ points.
I am having trouble visualizing the above construction. I can work everything out and e.g. construct a triangulation of $\Sigma_2$ that is preserved by $f$, but I cannot "see" $f$. Is there a picture of this diffeomorphism somewhere, or at least a more visual way of understanding it?
With the assistance of Big Bird. Look carefully at one of his "feet", and imagine it spinning. Let me know if you need a few more hints...