We develop a method to estimate from data travel latency cost functions in multi-class transportation networks, which accommodate different types of vehicles with very different characteristics (e.g., cars and trucks). Leveraging our earlier work on inverse variational inequalities, we develop a data-driven approach to estimate the travel latency cost functions. Extensive numerical experiments using benchmark networks, ranging from moderate-sized to large-sized, demonstrate the effectiveness and efficiency of our approach. ; Comment: Preprint submitted to the 56th IEEE Conference on Decision and Control (2017)