With the recent production of polar molecules in the quantum regime, long-range dipolar interactions are expected to facilitate the understanding of strongly interacting many-body quantum systems and to realize lattice spin models for exploring quantum magnetism. We now report the first observation of dipolar interactions of polar molecules pinned in a three-dimensional optical lattice. We realize a lattice spin model where spin is encoded in rotational states of molecules that are prepared and probed by microwaves. This interaction arises from the resonant exchange of rotational angular momentum between two molecules and realizes a spin-exchange interaction. The dipolar interactions are apparent in the evolution of the spin coherence, where we observe clear oscillations in addition to an overall decay of the coherence. The frequency of these oscillations, the strong dependence of the spin coherence time on the lattice filling factor, and the effect of a multi-pulse sequence designed to reverse dynamics due to two-body exchange interactions all provide clear evidence of dipolar interactions. The results reported here comprise an initial exploration of the behavior of many-body spin models with direct, long-range spin interactions and lay the groundwork for future studies of many-body dynamics in spin lattices.