We study tensor perturbations around the most general cosmologically
symmetric backgrounds in a class of teleparallel gravity theories known as New
General Relativity. These theories comprise a one-parameter class, which is
fully consistent with observations at the post-Newtonian level, and which
contains the teleparallel equivalent of General Relativity as a special case.
We find that for a particular class of cosmological background geometries these
theories exhibit gravitational wave birefringence and dispersion, i.e., the
dispersion relation for gravitational waves depends on their polarization and
wave number. The strength of this effect is directly related to the spatial
curvature of the cosmological background and the parameter describing the
deviation of the theory from General Relativity. We discuss the possibility of
observing this effect in gravitational wave experiments.