Traveling at the Speed of Thought tries to untangle the current state of science with respect to gravitational waves. The approach is mostly historical, with a focus on what Einstein thought at various times, and how others reacted to his analyses. The book is also fairly recent (2007) and doesn't seem to have been eclipsed by new discoveries yet. (There are active experiments looking for physical evidence of gravity waves. The effect is expected to be minuscule, so proponents are bothered by lack of success so far.)'s
A lot of the story is driven by a referee's report on a paper Einstein and Rosen submitted to Physical Review in 1936. Einstein was apparently used to European deference to authority on submitted papers, and was so upset that an anonymous referee had been consulted that he sent it to an obscure journal. By the time it was published there, Einstein had changed his mind about the primary conclusions.
The primary question seems to be whether gravitational waves carry energy with them as they propagate. If they do, then their sources (black holes, for instance) ought to lose mass over time. If they don't carry energy, then we don't have a sufficient theory of what could be propagating.
Another question that has to be answered is how fast gravity travels, and what it is that moves. In waves in water, individual water molecules only move locally, while the wave can travel great distances. With electromagnetism, actual photons move from place to place, carrying the influence. Which kind of thing is gravity? In one case, we should try to identify the medium in which the disturbance propagates, in the other, we should be able to find the particles themselves.
I may be over-simplifying, but the skeptical viewpoint seems to be that symmetrical motion or ballistic motion of any isolated mass wouldn't radiate energy, since changes in trajectory are required to produce gravity waves. If the present crop of detectors fails to find anything, this may be the best interpretation. It's hard to reconcile this suggestion with the presence of supernovas and binary star collapses. Those seem like dynamic enough changes that they should result in a change in the gravitational field that would have to propagate at some finite velocity.
's book provides a good, general introduction to the area, without getting too technical. If you're interested in the history it would make sense to read it. If you're looking for more details on what is know, how the math works, or how to interpret the results from the detectors, you should probably look elsewhere.