This is a question I asked years ago in an astronomy course, but to which I never got a straight answer. Please feel free to correct me if any of my assumptions here stray from facts. It goes like this:
To any observer, at large enough scale the universe appears to be receding in accordance with Hubble's constant, which causes the light that reaches them from objects such as distant galaxies to experience a red shift. Technically, though, if the observer is standing on a planet, they are in orbit moving in a particular direction at any given time, and so the distant galaxies in the direction of that should have a(n extremely) slightly smaller red shift while those in the opposite direction should have one slightly more severe. This means that, relatively, the galaxies behind are receding at a faster rate from the observer than those in front.
If the observer were to possess instruments of sufficient precision to determine the discrepancy in the red shifts and the means to adjust their velocity in any necessary way, would it be possible for them to equalize the red shift in every direction and thus be motionless in respect to the expansion of the universe itself? The cosmic background radiation might be a good reference for measuring red shifts in every direction, since it should be equally distant everywhere and very, very red shifted.