We propose a new data-driven approach for addressing multi-stage stochastic linear optimization problems with unknown distributions. The approach consists of solving a robust optimization problem that is constructed from sample paths of the underlying stochastic process. We provide asymptotic bounds on the gap between the optimal costs of the robust optimization problem and the underlying stochastic problem as more sample paths are obtained, and we characterize cases in which this gap is equal to zero. To the best of our knowledge, this is the first data-driven approach for multi-stage stochastic linear optimization that offers asymptotic optimality guarantees when uncertainty is arbitrarily correlated across time. Finally, we develop approximation algorithms for the proposed approach by extending techniques from the robust optimization literature, and demonstrate their practical value through numerical experiments on stylized data-driven inventory management problems.