We discuss model-free analysis of multisubject or multisession FMRI data by extending the single-session probabilistic independent component analysis model (PICA; Beckmann and Smith, 2004. IEEE Trans. on Medical Imaging, 23 (2) 137-152) to higher dimensions. This results in a three-way decomposition that represents the different signals and artefacts present in the data in terms of their temporal, spatial, and subject-dependent variations. The technique is derived from and compared with parallel factor analysis (PARAFAC; Harshman and Lundy, 1984. In Research methods for multimode data analysis, chapter 5, pages 122-215. Praeger, New York). Using simulated data as well as data from multisession and multisubject FMRI studies we demonstrate that the tensor PICA approach is able to efficiently and accurately extract signals of interest in the spatial, temporal, and subject/session domain. The final decompositions improve upon PARAFAC results in terms of greater accuracy, reduced interference between the different estimated sources (reduced cross-talk), robustness (against deviations of the data from modeling assumptions and against overfitting), and computational speed. On real FMRI 'activation' data, the tensor PICA approach is able to extract plausible activation maps, time courses, and session/subject modes as well as provide a rich description of additional processes of interest such as image artefacts or secondary activation patterns. The resulting data decomposition gives simple and useful representations of multisubject/multisession FMRI data that can aid the interpretation and optimization of group FMRI studies beyond what can be achieved using model-based analysis techniques.