  Koecher in 1980 derived a method for obtaining identities for the Riemann
zeta function at odd positive integers, including a classical result for
$\zeta(3)$ due to Markov and rediscovered by Ap\'ery. In this paper we extend
Koecher's method to a very general setting and prove two more specific but
still rather general results. As applications we obtain infinite classes of
identities for alternating Euler sums, further Markov-Ap\'ery type identities,
and identities for even powers of $\pi$
