The long-wave, small-amplitude dynamics of obliquely propagating Alfvén waves is shown, using a reductive perturbative expansion, to be purely linear not only in one space dimension but also in the dispersionless limit in higher dimensions. Furthermore, in the context of multidimensional wave-train modulation, all the diffraction coefficients are found to tend to zero with the dispersion, while the non-linear terms in the envelope equation remain finite. In this ‘semiclassical’ limit, the envelope dynamics results in the formation of growing regions of finite-amplitude oscillations with a typical scale intermediate between the size of the wave packet and its wavelength.