A unique sink orientation (USO) is an orientation of the edges of a polytope in which every face contains a unique sink. For a product of simplices $Δ_{m-1} \times Δ_{n-1}$, Felsner, Gärtner and Tschirschnitz (2005) characterize USOs which are induced by linear functions as the USOs on a $(m \times n)$-grid that correspond to a two-colored arrangement of lines. We generalize some of their results to products $Δ^1 \times\cdots\times Δ^r$ of $r$ simplices, USOs on $r$-dimensional grids and $(r+1)$-signotopes.