In General Relativity, disturbances in the gravitational field (i.e. spacetime curvature) propagate at the speed of light as gravitational radiation, also known as gravity waves. Thus, GR solves the original action at a distance problem of Newtonian gravity.
Light is produced by moving around electric charges. Gravity waves are produced by moving anything around. When you wave your hand, you make gravity waves.
The catch is that gravity is very weak. It takes a lot of mass, e.g. the sun or the earth, to generate substantial spacetime curvature. Likewise, it takes moving around very big masses to generate gravity waves that we can hope to measure.
In Fact, gravity waves have not yet been detected. The first detectors, really gravity wave telescopes, with enough sensitivity have only recently been built.
COSMOS to HUMANITY, Spring 2008
Lecture 2 , slide 7