One property that remains anisotropic even in cubic crystals is their elastic deformation. As described here, a cubic crystal has three elastic constants, and these can be varied in ways that create different deformation characteristics along different directions.
This is because elastic deformation is a more complex physical phenomenon than other processes such as heat conduction. Thermal conductivity correlates one vector (heat flux) to another (temperature gradient) and thus is a second-order tensor. The only component of a second-order tensor that obeys the symmetry laws associated with heat conduction and also conforms with cubic symmetry is the fully isotropic component, which corresponds to equal conductivity in all directions. But the elastic deformation constant correlates two quantities that are already second-order tensors: stress and strain. So the deformation constant is a fourth-order tensor, which has more components than a second-order one.
When we work out the mathematics of this fourth-order tensor for the cubic case, we find that three independent components are allowed. Only two of these correspond to equal elastic responses in all directions; these may be expressed in terms of Young's Modulus and Poisson's ratio. The third independent component can impart different deformation characteristics, which show up as different Young's Modulus and Poisson's ratio, to (for instance) deformation in the  direction versus the  direction.
An even higher level of symmetry, then, is required to get a truly isotropic crystal with respect to elastic deformation ... and now we know it exists thanks to the rise of quasicrystals. An icosahedral quasicrystal, which must obey more symmetry relations in its properties than a mere cubic crystal, really does have equal values of constants such as Young's Modulus and Poisson's ratio in all directions.