Discovering materials that display a linear magnetoelectric (ME) effect at room temperature is a challenge. Such materials could facilitate devices based on the electric field control of magnetism. Here we present simple, chemically intuitive design rules to identify a class of bulk magnetoelectric materials based on the ``bicolor'' layering of $Pbnm$ ferrite perovskites, e.g., ${\mathrm{LaFeO}}_{3}/{\mathrm{LnFeO}}_{3}$ superlattices, Ln = lanthanide cation. We use first-principles density functional theory calculations to confirm these ideas. We elucidate the origin of this effect and show it is a general consequence of the layering of any bicolor $Pbnm$ perovskite superlattice in which the number of constituent layers are odd (leading to a form of hybrid improper ferroelectricity). Our calculations suggest that the ME effect in these superlattices is larger than that observed in the prototypical magnetoelectric materials ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{BiFeO}}_{3}$. Furthermore, in these proposed materials, the strength of the linear ME coupling increases with the magnitude of the induced spontaneous polarization which is controlled by the La/Ln cation radius mismatch. We use a simple mean field model to show that the proposed materials order magnetically above room temperature.