Ok, then it's a matter of the power of the radiation? If so, there is a radiation corresponding to lower temperatures than 3000K which is too "weak" for us to see?
On the epoch of last scattering, the universe became transparent and the typical CMB photon was "free" to travel the universe. The corresponding radiation was of a black body temperature of ~3000K.
My question is: after the last scattering, the universe was still hot and, I presume, emitting...