We propose a Bayesian approach to regression with a scalar response on vector and tensor covariates. Vectorization of the tensor prior to analysis fails to exploit the structure, often leading to poor estimation and predictive performance. We introduce a novel class of multiway shrinkage priors for tensor coefficients in the regression setting and present posterior consistency results under mild conditions. A computationally efficient Markov chain Monte Carlo algorithm is developed for posterior computation. Simulation studies illustrate substantial gains over existing tensor regression methods in terms of estimation and parameter inference. Our approach is further illustrated in a neuroimaging application.