Classical random Boolean networks (RBN) are not well suited to describe experimental data from time-course microarray, mainly because of the strict assumptions about the synchronicity of the regulatory mechanisms. In order to overcome this setback, a generalization of the RBN model is described and analyzed. Gene products (e.g., regulatory proteins) are introduced, with each one characterized by a specific decay time, thereby introducing a form of memory in the system. The dynamics of these networks is analyzed, and it is shown that the distribution of the decay times has a strong effect that can be adequately described and understood. The implications for the dynamical criticality of the networks are also discussed.