Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speed-up over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random walk architecture that provides such a speed-up. It will be shown that this algorithm performs an oracle search on a database of $N$ items with $O(\sqrt{N})$ calls to the oracle, yielding a speed-up similar to other quantum search algorithms. It appears that the quantum random walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.