Kernel machines are one of the most studied family of methods in machine learning. In the exact setting, training requires to instantiate the kernel matrix, thereby prohibiting their application to large-sampled data. One popular kernel approximation strategy which allows to tackle large-sampled data consists in interpolating product kernels on a set of grid-structured inducing points. However, since the number of model parameters increases exponentially with the dimensionality of the data, these methods are limited to small-dimensional datasets. In this work we lift this limitation entirely by placing inducing points on a grid and constraining the primal weights to be a low-rank Canonical Polyadic Decomposition. We derive a block coordinate descent algorithm that efficiently exploits grid-structured inducing points. The computational complexity of the algorithm scales linearly both in the number of samples and in the dimensionality of the data for any product kernel. We demonstrate the performance of our algorithm on large-scale and high-dimensional data, achieving state-of-the art results on a laptop computer. Our results show that grid-structured approaches can work in higher-dimensional problems.