Never though about that. Isn't that easy to use to disprove FE gravity theory?
I claim that the tidal effect between, say, the ground and the top of Mt. Everest, or the ground and an airplane, are not large enough to be measured without extremely sensitive equipment that it would be very difficult for us to acquire and to bring to those locations.
troubador has found information indicating that the Earth is moving with respect to a field of thermal radiation left over from the Big Bang known as the Cosmic Microwave Background Radiation. While his claims that this CMBR act as an absolute frame of reference are, I think, erroneous (as that would overthrough general relativity completely), measuring the redshift and blueshift of this radiation will indeed indicate the Earth's motion with respect to the CMBR, and can thus be used to determine whether we are accelerating with respect to the CMBR.
There are a few problems with this suggestion, the most obvious of which is that again it is very difficult for us (the members of this forum) to measure this redshift, and so we are forced to rely on the reports of other scientists.
The next is that the CMBR cannot be an absolute frame of reference, meaning it is just as appropriate to say that
it is moving with respect to us as to say that
we are moving with respect to it -- that's the essence of relativity.
The last problem is that the CMBR is radiation, which seems to invalidate it as a candidate for any frame of reference, whether absolute or relative. The problem with light is that it always moves the same speed, which means you can't measure your speed by looking at how fast light goes. Just to compare, if you're moving through the air, you
can measure your airspeed by seeing how fast
sound moves, since the speed of sound is constant through air of a given temperature and pressure. Thus if you emit some sound and never hear it, you must be moving away from the point of emission at at least the speed of sound. You can't do anything like that with light, since the light will always reach you at the speed of light.
One way to get information about your motion by looking at light is by measuring the redshift of the spectrum of that light -- I will not go into details here. The conclusion that you draw, however, is not that you are moving a certain speed with respect to the light, but that you are moving a certain speed
with respect to the source of the light. So really, to say we are moving with respect to the CMBR is really to say that we are moving with respect to the
source of the CMBR, and it is not clear to me what that source is.