Einstein's general theory of relativity predicts that moving massive bodies will send out gravitational waves. As they pass through the Earth, they will cause the lengths of objects to change very slightly and so should be detectable. The problem is that they will change the lengths of objects by only one part in 1020 (100 billion billion). That's about one billionth of the width of a typical atom for a one meter object!
Very few scientists doubt that they exist but the challenges in observing them are so great that they haven't been observed directly yet. (There have been indirect observations.)
During the week, the LIGO gravitational wave observatory was approved for the upgrade to Advanced LIGO that will give it the power to actually detect gravitational waves. Wasn't that what LIGO was for? Well, not exactly. The technology is so challenging to implement that LIGO needed to run at a lower sensitivity first to ensure that everything was operational and to learn more about how it would run.
The transition to Advanced LIGO will increase the sensitivity of the instruments by a factor of 10, enough to detect the incredibly faint signals of gravitational waves roughly once per week.
The new observatory will be ready for operation in 2014 but you can get involved in the preliminary data analysis through the Einstein@home project.
For more information about the Advanced LIGO funding, see the news story in New Scientist.

And if you want a completely different take on the project, you can see and read about a 7-year-old's impression of LIGO in a previous story from symmetry.
For a last piece of trivia, there is a difference in physics terminology between gravitational waves and gravity waves. Gravitational waves are the waves predicted by Einstein but gravity waves are the everyday kind of waves like ocean waves that occur in a fluid when gravity is present.