We theoretically study spin transport in two-or three-dimensional Rashba systems dynamically distorted by surface acoustic waves. The spin currents in the linear response to lattice distortion dynamics are calculated on the basis of a microscopic theory combined with local coordinate transformations. As a result, we find a mechanism of direct spin-current generation from lattice distortion not associated with a charge current or spin accumulation. Moreover, the in-plane helicity currents are generated by shear surface acoustic waves via the present mechanism. The generated helicity currents are not parallel to the vorticity of the lattice, and cannot be created with the conventional methods. Thus, our findings offer an alternative functionality of the conventional Rashba systems in the field of spintronics.