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Plasmon comments on Some recent evidence against the Big Bang - Less Wrong Discussion
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Why doesn't this just mean that we are moving w.r.t. the rest frame of the CMB? The signal is redshifted in the hemisphere we're moving away from, and blueshifted in the hemisphere we're moving towards, so it would look hotter in the hemisphere we're moving towards.
Yes, that is precisely correct. This is one anisotropy that you expect to see in any model, because it's a fact about us, not about the universe.
The higher modes, however, suggest that the Big Bang wasn't homogeneous and isotropic. That doesn't make it not a BIG BANG in a general sense, nor not the Big Bang in the technical sense. It just means that there was more going on than we knew about. We already knew that.
The anisotropy you speak of from motion does exist. When you extract such a dipole from the data and renormalize though (you can very precisely calculate what the effects of motion would be and fit the direction to the radiation we observe) an asymmetry remains along a different axis.