Contrary to popular belief, modern cosmology doesn't give the Big Bang a specific location. Instead, the Universe expanded from a zero size (if you believe in Relativity) or from a size dependent on Planck's constant (if you believe in Quantum Mechanics). The precise analogy is that of a balloon inflating, but imagine it starting from zero size. So all points originate at that singularity, and the Big Bang can be said to have happened at all points.

In particular, you get uniform background radiation of 3°K.