  A new family of discrete-time quantum walks (DTQWs) propagating on a regular
$(1+2)$D spacetime lattice is introduced. The continuous limit of these DTQWs
is shown to coincide with the dynamics of a Dirac fermion interacting with an
arbitrary relativistic gravitational field. This family is used to model the
influence of arbitrary linear gravitational waves (GWs) on DTQWs. Pure shear
GWs are studied in detail. On large spatial scales, pure shear GWs do not
modify the polarizations of the fermion eigen-modes, but rescale all energies
by a common factor. On smaller scales typically comparable to two or three
lattice steps, both polarizations and energies are modified in a non-trivial
way. This effect is particularly salient on interference patterns between the
fermion eigen-modes.
