Dependency grammar induction is the task of learning dependency syntax without annotated training data. Traditional graph-based models with global inference achieve state-of-the-art results on this task but they require $O(n^3)$ run time. Transition-based models enable faster inference with $O(n)$ time complexity, but their performance still lags behind. In this work, we propose a neural transition-based parser for dependency grammar induction, whose inference procedure utilizes rich neural features with $O(n)$ time complexity. We train the parser with an integration of variational inference, posterior regularization and variance reduction techniques. The resulting framework outperforms previous unsupervised transition-based dependency parsers and achieves performance comparable to graph-based models, both on the English Penn Treebank and on the Universal Dependency Treebank. In an empirical comparison, we show that our approach substantially increases parsing speed over graph-based models.