Along with the advances of statistical data collection worldwide, dynamic factor models have gained prominence in economics and finance when dealing with data rich environments. Although factor models have been typically applied to two-dimensional data, three-way array data sets are becoming increasingly available. Motivated by the tensor decomposition literature, we propose a dynamic factor model for three-way data. We show that this modeling strategy is flexible while remaining quite parsimonious, in sharp contrast with previous approaches. We discuss identification and put forward a set of identifying restrictions that enhance the interpretation of the model. We propose an estimation procedure based on maximum likelihood using the Expectation-Conditional Maximization algorithm and assess the finite sample properties of the estimator through a Monte Carlo study. In the empirical application, we apply the model to inflation data for nineteen euro area countries and fifty-five products covering the last two decades.