  We investigate the new observational constraints on $f(T)$ gravity that arise
from the effects of primordial gravitational waves (GWs) on the cosmic
microwave background (CMB) anisotropies and the BB spectrum. We first show that
on the GWs propagation in $f(T)$ gravity we obtain only an amplitude
modification and not a phase one, comparing to the case of general relativity
in the background of $\Lambda$CDM cosmology. Concerning primordial GWs we find
that the more the model departs from general relativity the larger is the GW
amplitude decay, and thus a possible future detection would bring the viable
$f(T)$ gravity models five orders of magnitude closer to $\Lambda$CDM cosmology
comparing to standard cosmological constraints. Additionally, we use the CLASS
code and both data from the Planck probe, as well as forecasts from the
near-future CORE collaboration, and we show that possible non-trivial
constraints on the tensor-to-scalar ratio would offer a clear signature of
$f(T)$ gravity. Finally, we discuss on the possibility to use the properties of
the GWs that arise from neutron stars mergers in order to extract additional
constrains on the theory.
