I know there are no preferred inertial frames from the perspactive of special relativity, and I assume the same is true from the perspective of General Relativity. However, if the same is cosmologically true, then it seems that the distant universe should look the same regardless of one's motion. For example, the cosmic microwave background should appear the same temperature whether we're floating in the Oort cloud or traveling at 95% of the speed of light with respect to the Solar System. Is that true?
No it isn't true.
Take something simpler to illustrate the point: a glowing cloud here on Earth. Relativity says there are no preferred frames as far as physical laws etc. are concerned, but nevertheless this cloud will be at rest in one frame and not in others. If you stand inside the cloud and are motionless in the rest frame of the cloud, then you will see isotropic radiation: the same wavelength in all directions. If you are inside the cloud and moving relative to its rest frame, then you will see blue-shifted light in one direction and red-shifted in another, so it is no longer isotropic.
This picture carries over to observations of the cosmic microwave background radiation. The cloud corresponds to the matter which emitted the radiation long ago. There is a further complicating factor of the cosmic expansion and gravity, but apart from that the picture carries over quite well. At any given location in the universe there is one frame in which the CMB is isotropic, and other frames in which it is not. Thus the CMB does single out a "preferred frame" as far as observations of the cosmos go. But this is no different from the way the walls and roof of a house single out a preferred frame for the inhabitants of the house: it is the one in which the house is at rest.
The Local Group of galaxies is moving at 600 kilometers per second relative to the frame in which the CMB in our neighbourhood would be isotropic. See e.g. How fast is Earth moving through the universe? for more info.