ABSTRACT We present a systematic shearing-box investigation of magnetorotational instability (MRI)-driven turbulence in a weakly collisional plasma by including the effects of an anisotropic pressure stress, i.e. anisotropic (Braginskii) viscosity. We constrain the pressure anisotropy (Δp) to lie within the stability bounds that would be otherwise imposed by kinetic microinstabilities. We explore a broad region of parameter space by considering different Reynolds numbers and magnetic-field configurations, including net vertical flux, net toroidal-vertical flux, and zero net flux. Remarkably, we find that the level of turbulence and angular-momentum transport are not greatly affected by large anisotropic viscosities: the Maxwell and Reynolds stresses do not differ much from the MHD result. Angular-momentum transport in Braginskii MHD still depends strongly on isotropic dissipation, e.g. the isotropic magnetic Prandtl number, even when the anisotropic viscosity is orders of magnitude larger than the isotropic diffusivities. Braginskii viscosity nevertheless changes the flow structure, rearranging the turbulence to largely counter the parallel rate of strain from the background shear. We also show that the volume-averaged pressure anisotropy and anisotropic viscous transport decrease with increasing isotropic Reynolds number (Re); e.g. in simulations with net vertical field, the ratio of anisotropic to Maxwell stress (αA/αM) decreases from ∼0.5 to ∼0.1 as we move from Re ∼ 103 to Re ∼ 104, while 〈4$π$Δp/B2〉 → 0. Anisotropic transport may thus become negligible at high Re. Anisotropic viscosity nevertheless becomes the dominant source of heating at large Re, accounting for ${\gtrsim } 50 {{\ \rm per\ cent}}$ of the plasma heating. We conclude by briefly discussing the implications of our results for radiatively inefficient accretion flows on to black holes.