  We study the eta invariants of compact flat spin manifolds of dimension n
with holonomy group cyclic of odd prime order p. We find explicit expressions
for the twisted and relative eta invariants and show that the reduced eta
invariant is always an integer, except in a single case, when p=n=3. We use the
expressions obtained to show that any such manifold is trivial in the
appropriate reduced equivariant spin bordism group.
