No, motion does have an effect on gravity. However, the effect is not as simple as using the Newtonian formula for gravity and replacing "mass" with "relativistic mass".
A close, but imperfect, analogy for what happens to the gravitational field of a moving mass is what happens to the electric field of a moving charge. The case of the moving charge is considerably easier to understand, however. See for example http://www.phys.ufl.edu/~rfield/PHY2...ativity_15.pdf
(and also *_14.pdf).
In words: the electric field becomes concentrated in the transverse direction. A similar effect occurs for gravity.
For more discussion see the following old thread: