Current refugee resettlement processes account for neither the preferences of refugees nor the priorities of hosting communities. We introduce a new framework for matching with multidimensional knapsack constraints that captures the (possibly multidimensional) sizes of refugee families and the capacities of communities. We propose four refugee resettlement mechanisms and two solution concepts that can be used in refugee resettlement matching under various institutional and informational constraints. Our theoretical results and simulations using refugee resettlement data suggest that preference-based matching mechanisms can improve match efficiency, respect priorities of communities, and incentivize refugees to report where they would prefer to settle.