  We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin
manifolds of signature $(p,q)$, considered as an unbounded operator in the
Hilbert space $L^2_\xi(S)$. The definition of $L^2_\xi(S)$ involves the choice
of a $p$-dimensional time-like subbundle $\xi\subset TM$. We establish a
sufficient criterion for the spectra of $D$ induced by two maximal time-like
subbundles $\xi_1,\xi_2\subset TM$ to be equal. If the base manifold $M$ is
compact, the spectrum does not depend on $\xi$ at all. We then proceed by
explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat
torus $\mathbb T^{p,q}$ and products of the form $\mathbb T^{1,1}\times F$ with
$F$ being an arbitrary compact, even-dimensional Riemannian spin manifold.
