However, one thing we can observe is the light from shortly after the Big Bang (about 300,000 years after). This "Cosmic Microwave Background Radiation" was discovered in the 1960s and has dimmed considerably over the last 14 billion years. It has the spectrum of a "black body" -- like the light you get off a hot piece of metal, like a lightbulb filament, except that this piece of metal is 2.725 Kelvins.
The COBE and WMAP satellites have observed that the light is slightly bluer in one direction in space (l = 264 degrees, b = 48 degrees) by about 3.4 milliKelvins and slightly redder in the opposite direction. We believe this is like the Doppler shift that happens when an ambulance drives by and shifts in pitch -- in other words, that the Solar System is flying through this background radiation, left over from the Big Bang, at a particular speed and direction.
We can calculate the velocity that must be:
Solving for v, we get v = 229 miles per second.
So, to sum up, in theory anybody in the universe who can measure the CMB precisely enough can agree on a "CMB rest frame" that is moving, relative to our Solar System, in the opposite direction from the galactic coordinates l = 264 degrees, b = 48 degrees, at the speed of 229 miles per second.