You've described a linearly polarized wave, which has no angular momentum (which is I think what you're asking about). You could have a wave with displacements in each of the directions at right angles to the propagation direction, and it can have angular momentum.
Let's let the displacements be:
x=A cos(wt-kz) and
Look at any particle at some z, e.g. the one at z=0. For it
x=A cos(wt) and y=A sin(wt).
That's simple circular motion.
For other types of transverse wave, such as electromagnetic, the angular momentum takes different forms. There are still linearly polarized waves with no angular momentum and circularly polarized ones with angular momentum.
For longitudinal waves, such as sound in air, the displacements are along the same direction the wave propagates. Nothing is going around, and there is no angular momentum.
(published on 04/05/2009)