Differential privacy is a rigorous privacy condition achieved by randomizing query answers. This paper develops efficient algorithms for answering multiple queries under differential privacy with low error. We pursue this goal by advancing a recent approach called the matrix mechanism, which generalizes standard differentially private mechanisms. This new mechanism works by first answering a different set of queries (a strategy) and then inferring the answers to the desired workload of queries. Although a few strategies are known to work well on specific workloads, finding the strategy which minimizes error on an arbitrary workload is intractable. We prove a new lower bound on the optimal error of this mechanism, and we propose an efficient algorithm that approaches this bound for a wide range of workloads.