We propose a tensor neural network ($t$-NN) framework that offers an exciting new paradigm for designing neural networks with multidimensional (tensor) data. Our network architecture is based on the $t$-product (Kilmer and Martin, 2011), an algebraic formulation to multiply tensors via circulant convolution. In this $t$-product algebra, we interpret tensors as $t$-linear operators analogous to matrices as linear operators, and hence our framework inherits mimetic matrix properties. To exemplify the elegant, matrix-mimetic algebraic structure of our $t$-NNs, we expand on recent work (Haber and Ruthotto, 2017) which interprets deep neural networks as discretizations of non-linear differential equations and introduces stable neural networks which promote superior generalization. Motivated by this dynamic framework, we introduce a stable $t$-NN which facilitates more rapid learning because of its reduced, more powerful parameterization. Through our high-dimensional design, we create a more compact parameter space and extract multidimensional correlations otherwise latent in traditional algorithms. We further generalize our $t$-NN framework to a family of tensor-tensor products (Kernfeld, Kilmer, and Aeron, 2015) which still induce a matrix-mimetic algebraic structure. Through numerical experiments on the MNIST and CIFAR-10 datasets, we demonstrate the more powerful parameterizations and improved generalizability of stable $t$-NNs.