  Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin
structure. In this paper we determine the spectrum of Dirac operators acting on
smooth sections of twisted spinor bundles of $M$, and we derive a formula for
the corresponding eta series. In the case of manifolds with holonomy group
$\Z_2^k$, we give a very simple expression for the multiplicities of
eigenvalues that allows to compute explicitly the $\eta$-series in terms of
values of Riemann-Hurwitz zeta functions, and the $\eta$-invariant. We give the
dimension of the space of harmonic spinors and characterize all
$\Z_2^k$-manifolds having asymmetric Dirac spectrum.
  Furthermore, we exhibit many examples of Dirac isospectral pairs of
$\Z_2^k$-manifolds which do not satisfy other types of isospectrality. In one
of the main examples, we construct a large family of Dirac isospectral compact
flat $n$-manifolds, pairwise non-homeomorphic to each other.
