In this work we describe the effect of the electron momentum relaxation time on the terahertz properties of periodically-patterned graphene plasmonic structures; more specifically, its effect on the quality of the characteristic plasmonic resonances. From analytical theory as well as numerical simulations, it is observed that the electron momentum relaxation time heavily affects the quality factor of these resonances. In general, for a given electron concentration level, the larger the electron momentum relaxation time the larger the quality factor at resonance.