Conditional random fields and other graphical models have achieved state of the art results in a variety of tasks such as coreference, relation extraction, data integration, and parsing. Increasingly, practitioners are using models with more complex structure---higher tree-width, larger fan-out, more features, and more data---rendering even approximate inference methods such as MCMC inefficient. In this paper we propose an alternative MCMC sampling scheme in which transition probabilities are approximated by sampling from the set of relevant factors. We demonstrate that our method converges more quickly than a traditional MCMC sampler for both marginal and MAP inference. In an author coreference task with over 5 million mentions, we achieve a 13 times speedup over regular MCMC inference.