We perform direct numerical simulations (DNS) of homogeneous turbulence subject to periodic shear – $S \,{=}\, S_{\hbox{\scriptsize\it max}} \sin (\omega t)$, where $\omega$ is the forcing frequency and $S_{\hbox{\scriptsize\it max}}$ is the maximum shear. The lattice Boltzmann method (LBM) is employed in our simulations and a periodic body force is introduced to produce the required shear. We find that the turbulence behaviour is a strong function of the forcing frequency. There exists a critical frequency – $\omega_{cr}/S_{\hbox{\scriptsize\it max}} \,{\approx}\, 0.5$ – at which the observed behaviour bifurcates. At lower forcing frequencies ($\omega \,{<}\, \omega_{cr}$), turbulence is sustained and the kinetic energy grows. At higher frequencies, the kinetic energy decays. It is shown that the phase difference between the applied strain and the Reynolds stress decreases monotonically from $\pi$ in the constant shear case to $\pi/2$ in very high frequency shear cases. As a result, the net turbulence production per cycle decreases with increasing frequency. In fact, at $\omega/S_{\hbox{\scriptsize\it max}} \,{\geq}\, 10$, decaying isotropic turbulence results are recovered. The frequency-dependence of anisotropy and Reynolds stress budget are also investigated in detail. It is shown that inviscid rapid distortion theory (RDT) does not capture the observed features: it predicts purely oscillatory behaviour at all forcing frequencies. Second moment closure models do predict growth at low frequencies and decay at high frequencies, but the critical frequency value is underestimated. The challenges posed by this flow to turbulence closure modelling are identified.