We study the solution of linear systems resulting from the discreitization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure and, thus, solving them can be both time- and computer memory-consuming. Under certain assumptions, we show that the solution of such linear systems can be approximated with a vector of low tensor rank. We then solve the linear systems using low rank preconditioned iterative solvers. Numerical experiments demonstrate that these low rank preconditioned solvers are effective.