Not only does your speed affect the amount of radiation that you receive, but this actually happens to the Earth and has been measured experimentally.
You say:
So basically in space, there is bound to be stray radiation, whether from the stars, or cosmic background, floating around right.
and the most obvious example of this is the cosmic microwave background that pervades the whole universe. If you're at rest with respect to the average distribution of matter in the universe then the CMB looks isotropic - that is, it looks the same in all directions. However if you're moving then the CMB is blue shifted in your direction of motion and red shifted in the opposite direction. That means the CMB gets hotter and more intense in your direction of motion and cooler and less intense in the opposite direction.
Because the Solar System is moving wrt the average matter distribution we see this effect in measurements of the CMB made by the the Planck mission. The CMB is very slightly hotter in our direction of motion. This effect is called the dipole anisotropy.
If you're interested in pursuing this further see the question How is the Plane of the Solar System oriented to the Sun's motion through space?
