I've mentioned earlier a very concise paper by Lev Okun that made a very clear argument on why the concept of relativistic mass is really not accurate. In it, it was argued that it is the concept of energy and momentum that actually make more sense, and more meaningful as far as conceptualization is concerned. Now comes a new paper that traces not only the physics of mass and energy, but also the historical development of the idea leading up to Einstein's realization that "relativistic mass" isn't accurate.
In the first paragraph of the introduction alone, one can see that this is highly consistent with Lev Okun's earlier assertion:
Einstein's first paper on relativity appeared when the concept of speed-dependent electromagnetic mass had already become a topic of considerable interest. He accept this idea but changed his mind after being confronted by a far more compelling insight. We will show that after reading Planck's 1906 article in which the concept of relativistic momentum was introduced, Einstein came to realize that it was the relativistic equations for energy and momentum that were primary. From that perspective, it became clear that the inertial mass m was invariant, and he never again spoke of mass as being speed dependent.
This is a very good paper, especially if one is interested in the historical development of the concept of "relativistic mass" and why it shouldn't be used.
 E. Hecht, Am. J. Phys. v.77, p.799 (2009).