Observed delays in the ground response to solar wind directional discontinuities have been explained as the result of larger than expected magnetosheath propagation times. Recently, Samsonov et al. (2017, https://doi.org/10.1002/2017GL075020) showed that the typical time for a southward interplanetary magnetic field (IMF) turning to propagate across the magnetosheath is 14 min. Here by using a combination of magnetohydrodynamic simulations, spacecraft observations, and analytic calculations, we study the dependence of the propagation time on solar wind parameters and near-magnetopause cutoff speed. Increases in the solar wind speed result in greater magnetosheath plasma flow velocities, decreases in the magnetosheath thickness and, as a result, decreases in the propagation time. Increases in the IMF strength result in increases in the magnetosheath thickness and increases in the propagation time. Both magnetohydrodynamic simulations and observations suggest that propagation times are slightly smaller for northward IMF turnings. Magnetosheath flow deceleration must be taken into account when predicting the arrival times of solar wind structures at the dayside magnetopause.