Owing to prominence as a diagnostic tool for probing the neural correlates of cognition, neuroimaging tensor data has been the focus of intense investigation. Although many supervised tensor learning approaches have been proposed, they either cannot capture the nonlinear relationships of tensor data or cannot preserve the complex multi-way structural information. In this paper, we propose a Multi-way Multi-level Kernel (MMK) model that can extract discriminative, nonlinear and structural preserving representations of tensor data. Specifically, we introduce a kernelized CP tensor factorization technique, which is equivalent to performing the low-rank tensor factorization in a possibly much higher dimensional space that is implicitly defined by the kernel function. We further employ a multi-way nonlinear feature mapping to derive the dual structural preserving kernels, which are used in conjunction with kernel machines (e.g., SVM). Extensive experiments on real-world neuroimages demonstrate that the proposed MMK method can effectively boost the classification performance on diverse brain disorders (i.e., Alzheimer's disease, ADHD, and HIV).