  The starting point of any general relativistic numerical simulation is a
solution of the Hamiltonian and momentum constraints that (ideally) represents
an astrophysically realistic scenario. We present a new method to produce
initial data sets for binary neutron stars with arbitrary spins and orbital
eccentricities. The method only provides approximate solutions to the
constraints. However, we show that the corresponding constraint violations
subside after a couple of orbits, becoming comparable to those found in
evolutions of standard conformally flat, helically symmetric binary initial
data. We evolve in time three data sets, corresponding to binaries with spins
aligned, zero and anti-aligned with the orbital angular momentum. These
simulations show the orbital "hang-up" effect previously seen in binary black
holes. Additionally, all three show orbital eccentricities up to one order of
magnitude smaller than those found in helically symmetric initial sets
evolutions.
