  Recently, inelastic neutron scattering (INS) experiments on the insulating
parent compounds of high-T_c materials were analyzed to extract the value of
the superexchange constant $J$. Starting point of the analysis was the
nearest-neighbor Heisenberg model. Motivated by recent ARPES experiments, we
consider the effects of a next-nearest neighbor hopping, $t'$ in the strong
coupling limit of the spin-density wave formalism, where it leads to an
antiferromagnetic exchange $J'>0$ between next-nearest neighbor spins. We show
that a finite $J'$ (a) changes the spin-wave velocity and thus the value of $J$
extracted from the INS data, yielding almost identical values for $J$ in
La_2CuO_4 and YBa_2Cu_3O_{6.15}, and (b) leads to a considerable change in the
high-frequency spin-dynamics. Finally, we calculate the 1/S quantum corrections
in the presence of $J'$ and discuss their relevance for the INS data.
