There is a very famous principle called ''Neumann's Principle'', which states: if a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements.
Now, you know that the admissible symmetries on each lattice differ. For example, a monoclinic system accommodates 2-fold rotation, whereas a hexagonal (trigonal) system has 6 (3)-fold rotational symmetry. Different symmetries of different lattices may, by virtue of Neumann's principle, cause their physical properties to be different. The way the symmetries affect the physical property also depends on the property you are looking at (it being a scalar, a vector, or a tensor), so there is not a straight answer to your OP. I invite you to consult Nye's magnificent book for details.