Multivariate observations over the items and across the subjects with longitudinal and cross-sectional dependence naturally form a stochastic tensor data structure. Several types of changes in tensor means are considered. A class of changepoint detection methods is investigated. These procedures do not require training data and, moreover, are completely distribution-free and tuning-parameter-free. We propose SVD-bootstrap superstructure that overcomes the computational curse of dimensionality without any loss of information. The empirical properties of the detection technique are investigated through a simulation study. The fully data-driven test is applied to real-world data from EEG.