Is there a Cosmic Gravitational Background Radiation (CGBR)?

Yes, there is a predicted cosmic gravitational wave background; an expanding universe is essentially transparent to gravitational waves once they are produced.

There are a number of scenarios in which GWs might be produced - these are listed in the eLISA white paper and include a transition from the inflationary epoch to the hot big bang; but for realistically detectable strain values, an interferometer needs to be sensitive to milli-Hz or lower.

This means that the gravitational waves have extremely long wavelengths and milli-Hz frequencies are masked by seismic activity on Earth. Thus one needs an interferometer with very long arms in space.

Detecting the cosmic gravitational wave background is one of the things the space-based eLISA interferometer may be able to do with its proposed 1 million km arms.

See http://www.ligo.org/science/GW-Stochastic.php