Everything on the universe would move in this exact direction it was moving at the instant gravity stopped.
Imagine a few galaxies, separated by say 20 000 light years from each other. When gravity is present, they pull inwards towards each other. If gravity was turned off, they would still continue on the path they were on, because there is nothing to slow them down.
But, if there is no gravity, the galaxies themselves would start to fall apart, with each star continuing in a straight line, rather than it's previous curved course around the galaxy centre of mass.
Since it is gravity that binds planets in shape, when gravity disappears, planets will lose their shape are turn into dust clouds.
As far as planets go, there are other forces, such as the electromagnetic force, which is far stronger than gravity at close range, so atoms and molecules (and us) would still stay together as units, although moving away from Earth , as John refers to in his comments above.
For the Earth, we would lose the moon, it would head off at a tangent, but planets would not "fall to dust".
EDIT Roland's answer regarding the Earth, is correct (and mine is wrong) but my guess is we would go back to the state the primordial solar system was in, (leaving the sun out of it , just for the moment, as we would be doomed anyway), with large chunks of rocks moving around in totally chaotic orbits. Although the Earth would fall apart, it would not be to the level of dust. END EDIT