In this paper we consider the finite-time stability problem for discrete-time linear systems. Differently from previous papers, the stability analysis is performed with the aid of polyhedral Lyapunov functions rather than using the classical quadratic Lyapunov functions. In this way we are able to deal with more realistic constraints on the state variables; indeed, in a way which is naturally compatible with polyhedral functions, we assume that the sets to which the state variables must belong to in order to satisfy the finite-time stability requirement are boxes (or more in general polytopes) rather than ellipsoids. The main result, derived by using polyhedral Lyapunov functions, is a sufficient condition for finite-time stability of discrete-time linear systems. Some examples are presented to illustrate the advantages of the proposed methodology over existing methods.