> But I suspect we'd still be able to detect which "direction" the wave was traveling in. Time dilation would be a matter of simple velocity causing a slowdown, it would be velocity in a certain direction causing a slowdown, and velocity in another direction causing a speed-up.

Technically, perhaps we would. But picture yourself as a tiny little point on a vast plane. Down from above (A direction you cannot travel in; you're restricted to two dimensions) shines a light which covers the entire plane. Technically, as you draw away from the point-source, it gets dimmer, but your field of movement is so restricted that you cannot ever draw away far enough to cause a serious impact on the illumination.