Waves are generally classified as either transverse or longitudinal depending on the they way the propagated quantity is oriented with respect to the direction of propagation. Then what is a gravitational wave? It doesn't make sense to me that a disturbance in the curvature of spacetime has a "direction", so I would say they're neither, much like a wave packet in quantum mechanics.
Answer
Gravitational waves are transverse waves but they are not dipole transverse waves like most electromagnetic waves, they are quadrupole waves. They simultaneously squeeze and stretch matter in two perpendicular directions. Gravitational waves definitely propagate in a given direction but the effect that they have on matter is completely perpendicular to the direction of motion. Below is a picture of what the metric of a passing wave does to space (the wave traveling is perpendicular to the screen). If you imagine a free particle sitting at each grid intersection point, the particle would move sinusoidally right along with the grid:
This diagram is from this paper
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