We consider an intermediary's problem of dynamically matching demand and supply of heterogeneous types in a periodic-review fashion. More specifically, there are two disjoint sets of demand and supply types. There is a reward associated with each possible matching of a demand type and a supply type. In each period, demand and supply of various types arrive in random quantities. The platform's problem is to decide on the optimal matching policy to maximize the total discounted rewards minus costs, given that unmatched demand and supply will incur waiting or holding costs, and will be carried over to the next period with abandonments. This problem applies to many emerging settings in the sharing economy and also includes many existing problems, e.g., assignment/transportation problems, as special cases. For this dynamic matching problem, we provide sufficient conditions (which we call "modified Monge conditions") only on matching rewards such that the optimal matching policy follows a priority hierarchy among possible matching pairs: if some pair of demand and supply types is not matched as much as possible, all pairs that have strictly lower priority down the hierarchy should not be matched. This result is obtained by a generalization of the classic augmenting path approach to the stochastic dynamic program. As a result of the priority property, the optimal matching policy boils down to a match-down-to threshold structure when considering a specific pair of demand and supply types, along the priority hierarchy.