  We study a chain of infinitely many particles coupled by nonlinear springs,
obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) -
V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that
this chain carries exact spatially periodic travelling waves whose profile is
asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The
discrete waves have three interesting features: (1) being exact travelling
waves they keep their shape for infinite time, rather than just up to a
timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis,
(2) unlike solitary waves they carry a nonzero amount of energy per particle,
(3) analogous behaviour of their KdV continuum counterparts suggests long-time
stability properties under nonlinear interaction with each other. Connections
with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an
adaptation of the renormalization approach of Friesecke and Pego (1999) to a
periodic setting and the spectral theory of the periodic Schr\"odinger operator
with KdV cnoidal wave potential.
