The cosmic background radiation was always with us, it is not reaching us now. It just became cooler and cooler as time went on.
One has to understand that when we are talking big bang and general relativity we are talking of the universe starting from one (x,y,z,t) point and as time goes on, expanding. This means that all (x,y,z) points of our universe trace back to that one point singularity that went "bang".
Envision the two dimensional surface of a balloon, as shown in the wiki link. At time=0 the balloon is one point, call it r=0. As it expands the points on its surface start receding from each other, and all points on that surface were at r=0 at t=0. Their neighborhood expands as the balloon blows up, and this means the electromagnetic radiation that started in the earth's neighborhood hot, cools due to the expansion and becomes the Cosmic Microwave Background. That is what I mean it is not coming from anywhere, it just is.