Compressive learning (CL) is an emerging topic that combines signal acquisition via compressive sensing (CS) and machine learning to perform inference tasks directly on a small number of measurements. Many data modalities naturally have a multidimensional or tensorial format, with each dimension or tensor mode representing different features such as the spatial and temporal information in video sequences or the spatial and spectral information in hyperspectral images. However, in existing CL frameworks, the CS component utilizes either random or learned linear projection on the vectorized signal to perform signal acquisition, thus discarding the multidimensional structure of the signals. In this article, we propose multilinear CL (MCL), a framework that takes into account the tensorial nature of multidimensional signals in the acquisition step and builds the subsequent inference model on the structurally sensed measurements. Our theoretical complexity analysis shows that the proposed framework is more efficient compared to its vector-based counterpart in both memory and computation requirement. With extensive experiments, we also empirically show that our MCL framework outperforms the vector-based framework in object classification and face recognition tasks, and scales favorably when the dimensionalities of the original signals increase, making it highly efficient for high-dimensional multidimensional signals.