In this paper we consider the state reconstruction problem with Boolean measurements for a genetic network governed by a piecewise affine (PWA) model. A Luenberger-like observer is proposed and we undertake an investigation of its convergence rate. The approach of Filippov is used to define the solution on the surface of discontinuity. In particular sliding modes may occur in some cases and this leads to finite-time convergence for the observer. A transition graph is given for the coupled observer-nominal system in general case. To minimize the convergence time, different convergence scenarios are discussed for optimizing the choice of initial condition of the observer.