Memristor crossbars are capable of implementing learning algorithms in a much more energy and area efficient manner compared to traditional systems. However, the programmable nature of memristor crossbars must first be explored on a smaller scale to see which memristor device structures are most suitable for applications in reconfigurable computing. In this paper, we demonstrate the programmability of memristor devices with filamentary switching based on LiNbO₃, a new resistive switching oxide. We show that a range of resistance values can be set within these memristor devices using a pulse train for programming. We also show that a neuromorphic crossbar containing eight memristors was capable of correctly implementing an OR function. This work demonstrates that lithium niobate memristors are strong candidates for use in neuromorphic computing.