Nope. Mass is a form of energy. (Your E = mc^2 and the like.)
Of the other forms of energy, there are two types.
The first is potential energy in a field (be it from the strong nuclear force, weak nf, electromagnetic or gravitational), the interaction taking place by carrier particles (eg. the photon, or the graviton).
The second is kinetic energy, which is from movement relative to the rest frame, and so is a bit more ambiguous.
ps. I'm assuming the graviton exists.