We present a theoretical description of the small-signal transient response of polariton laser diodes (pol-LDs) based on simplified coupled rate equations describing the exciton reservoir and the ground-state polariton populations. The analytic expressions derived for two pumping geometries, which are valid for all inorganic semiconductors suitable for the realization of pol-LDs, are compared to exact numerical calculations performed for the specific case of GaN-based devices. The two approaches show excellent agreement provided the current step transient remains within the small-signal limit. We report that the temporal attenuation of the envelopes of the oscillations matches half the value of the damping factor (gamma(d)) of the pol-LDs, which is proportional to the square of the oscillation relaxation resonance frequency. An explicit expression for the dependence of d g on both the exciton-photon detuning and the driving current (equivalently the optical pump power) is also obtained. In a further step, we derive the expression for the turn-on delay (t(d)) associated with the build-up of the exciton reservoir population up to its threshold value before coherent light emission occurs. We show that td has the same functional form for the two pumping geometries. It is equal to the effective exciton lifetime (tau(xeff)) weighted by a logarithmic dependence on the initial and final driving currents. In addition, tau(xeff) t is shown to be approximately equal to the exciton lifetime, which proves to be the main parameter governing the build-up of polariton lasing/condensation. Beyond electrically driven polariton lasers, we highlight that the temporal shape of the transients could also be easily tested by monitoring the time dependence of the output power of optically pumped polariton lasers subjected to a sudden increase in the continuous wave pump power within the small-signal limit.