In this paper, we consider finite-horizon predictive control of linear stochastic systems with chance constraints where the admissible region is a convex polytope. For this problem, we present a novel solution approach based on box approximations. The key notion of our approach consists of two steps. First, we apply a linear operation to the joint state probability density function such that its covariance is transformed into an identity matrix. This operation also defines the transformation of the state space and, therefore, of the admissible polytope. Second, we approximate the admissible region from the inside using axis-aligned boxes. By doing so, we obtain a conservative approximation of the constraint violation probability virtually in closed form (the expression contains Gaussian error functions). The presented control approach is demonstrated in a numerical example.