Linear filtering methods are well-known and have been successfully applied to system identification and equalization problems. However, when high-dimensional systems are modeled, these methods often perform unsatisfactorily due to their slow convergence and to the high number of parameters to estimate, which brings high computational and storage complexities. To cope with these difficulties, the assumption of a low-rank impulse response was recently exploited to derive computationally efficient adaptive tensor filtering methods. However, existing approaches either model the impulse response as a low-rank matrix or a rank-1 tensor. While the former choice can only bring a limited complexity reduction, the latter relies on a strong assumption that is too restrictive for many systems of interest. In this work, we propose an adaptive filtering approach that models the impulse response more generally as a low-rank tensor, with a rank possibly higher than one and order possibly higher than two. This approach is suitable for problems involving the identification of a high-dimensional system whose impulse response has a multilinear low-rank structure. It can overcome the curse of dimensionality by taking advantage of such a structure, which allows breaking a large system identification problem into several smaller ones. Simulation results compare the proposed algorithms with existing ones in terms of their performance, computational complexity, and memory demands. In particular, these results show that when the target impulse response has a tensor structure, our approach can achieve a faster convergence and lower steady-state error than standard LMS while having reduced memory complexity in comparison with the matrix-based approach.