Yes and no. Yes, you can in fact do measurements to determine whether you're moving with respect to the frame in which the cosmic microwave background exhibits no Doppler shift. But while finding this frame of reference might be interesting, it is still not an absolute frame of reference. That is, there's no difference in the laws of motion whether you're in a CMB-Doppler-shifted frame or a CMB-Doppler-unshifted frame. That is the more fundamental point. So we might decide to call the CMB-Doppler-unshifted frame a "standard" frame of reference, but it is still not an "absolute" frame of reference.
Incidentally, my understanding is that we measure very little Doppler shift in the CMB in our galaxy. However, even though we can measure very significant Doppler shifts in the light from distant galaxies, observers in these galaxies would also measure very little Doppler shift in the CMB. This is because the universe is expanding more or less symmetrically, so the "horizon" from which the CMB comes is essentially the same distance away no matter where you are. So an observer on Earth and an observer in a distant galaxy could both claim to be at rest with respect to the CMB standard frame, yet still be moving at relativistic speeds with respect to each other!