Summary: We give polynomial-time algorithms that can take a graph $G$ with a given combinatorial embedding on an orientable surface $S$ of genus $g$ and produce a planar drawing of $G$ in $\bold R^2$, with a bounding face defined by a polygonal schema $\cal P$ for $\cal S$. Our drawings are planar, but they allow for multiple copies of vertices and edges on $\cal P$’s boundary, which is a common way of visualizing higher-genus graphs in the plane. However, unlike traditional approaches the copies of the vertices might not be in perfect alignment but we guarantee that their order along the boundary is still preserved. Our drawings can be defined with respect to either a canonical polygonal schema or a polygonal cutset schema, which provides an interesting tradeoff, since canonical schemas have fewer sides, and have a nice topological structure, but they can have many more repeated vertices and edges than general polygonal cutsets. As a side note, we show that it is NP-complete to determine whether a given graph embedded in a genus-$g$ surface has a set of $2g$ fundamental cycles with vertex-disjoint interiors, which would be desirable from a graph-drawing perspective.