Do particles in a sound wave ever move transversally? I'm trying to visualize how sound waves work and I was curious about something. 
So sound moves in longitudinal waves, which I think I understand.
There is a really good khanacademy video explaining this.
In the video the air particles are depicted as moving from left to right (longitudinally). But do they ever move up and down even a little bit? Are they on some kind of fixed axis where they can only move longitudinally and never move even the slightest bit in a vertical direction?
 A: That video is very poor in one aspect: particles in the sound field doesn't move "horizontally" nor "vertically". Really, the proper word is "longitudinal motion" and you are in fact asking about "transversal motion".
In basic description, the air is considered to be an ideal fluid. Therefore no shear stress is possible and hence no transversal motion as well.
In fact, there is a (little) viscosity of air, so yes, it's generally possible for particles to move a bit transversal as well. It could be observed especially in boundary layers on solid bodies. Of course, inside or on top of solid bodies or highly-viscous fluids, the transversal motion combined with longitudinal is natural.
A: If you have any kind of solid material, it will become a little bit thicker as you compress it, and thinner as your stretch it. This means that a "one dimensional" wave traveling longitudinally down a rod will in fact cause some lateral motion. The ratio of displacements in the perpendicular direction is obtained from the strain (relative displacement of adjacent particles - how much local deformation there is) and the Poisson ratio of the material (a value between -1 and +0.5) that describes how much longitudinal strain turns into lateral strain.
So yes, it does move "the slightest bit".
A: If we assume that you're talking about a broad sound wave traveling through a large, homogenous medium (air, water, rock) then normally, no: the pressure waves that are sound involve only motion along the direction of the sound's travel. You can wonder if the higher-pressure zones would tend to push the particles of the medium sideways, but remember that there's a broad area of higher pressure all traveling together, so that the only "escape" for the compressed particles is to continue the motion of the sound wave.
If you instead have a thin beam of sound, then it's likely that you'll get some form of lateral movement at the edges (probably related to the concept of diffraction, as in waves in the ocean diffracting around a point of land).
If you start dealing with non-homogenous media, then all bets are off.
