Can the accumulation of gravitational wave energy explain Dark Energy? Gravitational waves do not dissipate.  How significant is this accumulation of gravitational waves throughout the entire history of the universe? How much energy is that accumulation and how does that compare to the density of "dark energy"?  Could they be  one and the same?
 A: Unfortunately, the only paper I know of offhand that addresses the issue has probably been superseded by research in the years since its publication;  still, their estimates are probably correct to within an order of magnitude or two.

Masataka Fukugita & P. J. E. Peebles, The Cosmic Energy Inventory. Astrophys. J. 616: 643–668 (2004).

Their estimates indicate that the energy density of gravitational waves is a negligible fraction of the total energy of the Universe.  There are two types of gravitational waves they consider:  primeval gravitational waves left over from the Big Bang, and gravitational radiation from things like stellar collapse, black hole mergers, etc.
Primeval gravitational waves
For primeval gravitational waves, Fukugita & Peebles argue that they comprise less than one part in 1010 of the energy density of the Universe:

The density parameter associated with gravitational waves with wavelengths on the order of the Hubble length is $\Omega_g \sim \delta^2$, and the absence of an appreciable effect of gravitational waves on the anisotropy of the 3 K thermal cosmic background radiation indicates $\delta \lesssim 10^{−5}$.

In the 13 years since this paper was published, there have been several attempts to detect these waves, including one well-publicized false signal that evaporated on further analysis.  I don't know what the current bound is on primeval gravitational waves, but I suspect that it could now be pushed several orders of magnitude lower than Fukugita & Peeble's estimate.
Gravitational radiation
Fukugita & Peebles consider the graviational radiation from two sources:  stellar-mass binaries and the formation of black holes.  For stellar-mass binaries, they estimate that the energy is about one part in 109 ± 1 of the total energy of the Universe;  for massive black holes, their estimate is about one part in 107 or 108.
Again, I must mention that this paper was published 13 years ago.  Physicists' understanding of gravitational radiation has advanced quite a lot since then;  if I remember my history right, numerical simulations of black hole mergers couldn't even complete a full orbit until 2005 or so.  And, of course, two or three black hole merger events were detected by LIGO about 18 months ago.  All of these things probably allow these estimates to be refined further.
Finally, you ask about dark energy.  There is still a significant ongoing debate as to whether small-scale "ripples" in gravity could act like dark energy.  (The precise term they use is gravitational inhomogeneities;  these inhomogeneities would include not just gravitational waves, but also the gravitational wells of things like stars & galaxies.  Some physicists (most notably Rocky Kolb and his collaborators) say that it could, while others (most notably Robert Wald and his collaborators) say that it couldn't.  I was the doctoral student of one of the above physicists, but I have not looked at the papers in detail;  so I probably can't provide an impartial opinion as to whether this idea could work.  
