We develop a Bayesian approach to predict a continuous or binary outcome from data that are collected from multiple sources with a multi-way (i.e.. multidimensional tensor) structure. As a motivating example we consider molecular data from multiple 'omics sources, each measured over multiple developmental time points, as predictors of early-life iron deficiency (ID) in a rhesus monkey model. We use a linear model with a low-rank structure on the coefficients to capture multi-way dependence and model the variance of the coefficients separately across each source to infer their relative contributions. Conjugate priors facilitate an efficient Gibbs sampling algorithm for posterior inference, assuming a continuous outcome with normal errors or a binary outcome with a probit link. Simulations demonstrate that our model performs as expected in terms of misclassification rates and correlation of estimated coefficients with true coefficients, with large gains in performance by incorporating multi-way structure and modest gains when accounting for differing signal sizes across the different sources. Moreover, it provides robust classification of ID monkeys for our motivating application. Software in the form of R code is available at https://github.com/BiostatsKim/BayesMSMW .