The characteristics of steady granular flow in quasi-two-dimensional rotating tumblers have been thoroughly investigated and are fairly well understood. However, unsteady time-varying flow has not been studied in detail. The linear response of granular flow in quasi-two-dimensional rotating tumblers is presented for periodic forcing protocols via sinusoidal variation in the rotational speed of the tumbler and for step changes in rotational speed. Variations in the tumbler radius, particle size, and forcing frequency are explored. Similarities to steady flow include the fastest flow occurring at the free surface of the flowing layer and an instantaneous approximately linear velocity profile through the depth. The flowing layer depth varies by 2-5 particle diameters between minimum and maximum rotation rates. However, unsteady forcing also causes the flow to exhibit dynamic properties. For periodic rotational speeds, the phase lag of the flowing layer depth increases linearly with increasing input forcing frequency up to nearly 2.0 rad over 0-20 cycles per tumbler revolution. The amplitude responses of the velocity and shear rate show a resonance behavior unique to the system level parameters. The phase lag of all flow properties appears to be related to the number of particle contacts from the edge of the rotating tumbler. Characterization via step changes in rotational speed shows dynamic properties of overshoot (up to 35%) and rise times on the order of 0.2-0.7 s. The results suggest that the unsteady granular flow analysis may be beneficial for characterizing the "flowability" and "rheology" of granular materials based on particle size, moisture content, or other properties.