Let $Z$ be a variety and $A$ an elliptic curve over the function field of $Z$. I. Dolgachev and M. Gross define the \emph{geometric Shafarevich-Tate group} of $A$ over $Z$ to classify the set of isomorphism classes of principal homogeneous spaces for $A$ which are locally trivial in the \'etale topology. In joint work with Chad Schoen, we describe how to compute the Shafarevich-Tate group when $A$ is the generic fiber of a class of elliptic threefolds and $Z$ is the base. We also obtain results on the Brauer groups of such threefolds.