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As you may know, the linearized relativity theory (or gravitoelectromagnetism), obeys equations analogous to Maxwell's equations in electrodynamics. These equations in ED result in the troubling phenomenon of radiation reaction force when the accelerating charge sort of interacts with its own EM radiation field. So, I am asking, is there a similar phenomenon in relativity theory that mass interacts with its own gravitoelectromagnetic radiation field?

Certainly there must be, at least in the linearized gravity, since the mathematics is the same. I guess the situation is solved in general relativity somehow? Finally, is the phenomenon connected to the origins of inertia?

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There is absolutely a gravitational radiation reaction and solving for it is one of the very active fields in classical relativity theory at present. Basically, particles with nontrivial masses distort the spacetime around them; this causes them to not move on geodesics of the "background" spacetime (the spacetime that one would have found had the secondary particle not been there) but instead on some accelerated path. Solving for that path "self-consistently" at all orders is the holy grail of this field. This turns out to be very difficult.

Presently, the so-called "self-force" is understood to first order in the Schwarzschild spacetime. It will perhaps soon be understood to second order there and to first order in Kerr. The calculations are very challenging.

The self-force is of somewhat more practical interest than the electromagnetic radiation reaction because it is responsible for the inspiral and ultimate merger of binary black holes, which is the most important source of gravitational radiation. It turns out to have two components: a "conservative" one which respects the symmetries of the background spacetime, and a "dissipative" one which roughly accounts for the energy lost to gravitational radiation.

Understanding both is necessary to model gravitational wave sources, particularly in the "extreme high mass ratio limit" where $M_1 ~10^7$ solar masses and $M_2 ~ 1$ solar mass. This should realistically describe the case of, say, a stellar mass black hole merging with one of the supermassive black holes at the centre of a galaxy.

Here is a review: http://relativity.livingreviews.org/Articles/lrr-2011-7/

EDIT: The gravitational self-force is not responsible for inertia. It is a very weak effect normally.

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