The linear viscoelasticity of a dilute suspension of active (self-propelled) rigid spheroidal particles is calculated under a small-amplitude oscillatory shear (SAOS) deformation. The imposed shear acts to drive the microstructure of the suspension, as parameterized by the orientational probability distribution function, out of equilibrium. The microstructure relaxes via two independent mechanisms: rotational Brownian motion and correlated tumbling; the combination of which results in an increased rate of stress relaxation, relative to a suspension that relaxes solely by either mechanism. We explicitly calculate the non-equilibrium orientational microstructure due to the SAOS deformation, rotational diffusion, and tumbling. From this, we determine the linear viscoelasticity of the suspension from the orientationally averaged stresslet, which arises from the imposed flow, rotational diffusion, and particle activity (self-propulsion). Next, we demonstrate that a modified Cox-Merz rule is applicable to a dilute, active suspension via a comparison of our linear viscoelasticity results to a theoretical prediction of the steady shear viscosity of active, slender rods (Saintillan, Exp Mech 50(9) 1275-1281, 2010). Finally, through a comparison of our results to experiments on Escherichia coli (Lpez et al., Phys Rev Lett 115(2) 028, 301, 2015), we show that the linear viscoelasticity of an active suspension can be utilized to determine the mechanism of self-propulsion (i.e., pusher or puller), and estimate the strength of self-propulsion and correlation between tumbling events.