We present a computational framework for automatic synthesis of a feedback control strategy for a discrete-time piecewise affine (PWA) system from a specification given as a linear temporal logic (LTL) formula over an arbitrary set of linear predicates in the system's state variables. Our approach consists of two main steps. First, by defining appropriate partitions for its state and input spaces, we construct a finite abstraction of the PWA system in the form of a control transition system. Second, by leveraging ideas and techniques from LTL model checking and Rabin games, we develop an algorithm to generate a control strategy for the finite abstraction. While provably correct and robust to state measurements and small perturbations in the applied inputs, the overall procedure is conservative and expensive. The proposed algorithms have been implemented as a software package and made available for download. Illustrative examples are included.