We have studied the growth of a modulated (2,2) antiphase ordering in the two-dimensional anisotropic next-nearest-neighbor Ising model with Kawasaki dynamics using Monte Carlo methods. We have quenched systems of various sizes to several different quench temperatures below the phase-transition line for different values of the anisotropy parameter α. Both the domain growth and the structure factor are found to evolve in a strongly anisotropic manner. As well, the structure factor is shown to satisfy a generalized anisotropic scaling to a good degree of accuracy. We discuss in some detail both the energetics and the dynamical role of the different types of domain walls and excitations found in the system, as well as the effect of a wetting transition on the domain growth. We also point out that the growth law in our model is not accurately characterized by a simple power law in the time regimes studied.