We propose a novel data-driven framework for algorithmic recourse that offers users interventions to change their predicted outcome. Existing approaches to compute recourse find a set of points that satisfy some desiderata -- e.g. an intervention in the underlying causal graph, or minimizing a cost function. Satisfying these criteria, however, requires extensive knowledge of the underlying model structure, often an unrealistic amount of information in several domains. We propose a data-driven, computationally efficient approach to computing algorithmic recourse. We do so by suggesting directions in the data manifold that users can take to change their predicted outcome. We present Stepwise Explainable Paths (StEP), an axiomatically justified framework to compute direction-based algorithmic recourse. We offer a thorough empirical and theoretical investigation of StEP. StEP offers provable privacy and robustness guarantees, and outperforms the state-of-the-art on several established recourse desiderata.