We present the spectral and temporal radiative signatures expected within the "supercritical pile" model of gamma-ray bursts (GRBs). This model is motivated by the need for a process that provides the dissipation necessary in GRBs and presents a well-defined scheme for converting the energy stored in the relativistic protons of the relativistic blast waves (RBWs) associated with GRBs into radiation; at the same time, it leads to spectra that exhibit a peak in the burst νF_ν distribution at an energy E_peak 1 MeV in the observer's frame, in agreement with observations and largely independent of the Lorentz factor Γ of the associated relativistic outflow. Furthermore, this scheme does not require (but does not preclude) acceleration of particles at the shock other than that provided by the isotropization of the flow bulk kinetic energy in the RBW frame. In the present paper we model in detail the evolution of protons, electrons, and photons from a RBW to produce detailed spectra of the prompt GRB phase as a function of time from across a very broad range in frequency, spanning roughly 4 log Γ decades. The model spectra are in general agreement with observations and provide a means for delineating the model parameters through direct comparison with trends observed in GRB properties.