Needless to say, traditional Christians should have no objection to a contemplative practice that aims to assist them in becoming aware, or more fully aware, of the apophatic presence of God. The “traditionalist Catholic” criticism of the Centering Prayer which you mention, namely, that it is “too much like” a Buddhist quest for nirvana, is entirely beside the point.
That said, I do have some serious reservations about the Centering Prayer. For one thing, the presentations I’ve heard on this subject—notably a lecture some years ago by Father Thomas Keating—seemed to me to place far too much emphasis on the method’s psychological benefits. Such an emphasis can very easily lead the uninformed inquirer into thinking that spirituality is little more than a matter of relaxing and “letting God take over”. Serious seekers must realize that no transformative practice, or at least none worthy of the name, is going to be easy. There will always be a certain dimension of struggle and effort, a battle with habits and with what the Christian East calls the “passions”. Have the Centering proponents given this fact quite the attention it deserves?
There’s another, more problematic issue, however. Am I not right that practitioners of the Centering Prayer are told to choose their own “sacred word”—be it “love” or “happiness” or something equally anodyne—as a support for their “intentions”? You are right, of course, that their practice is outwardly similar in some respects to Hesychasm’s Prayer of the Heart. But in that tradition, it would be entirely out of the question for a person to make such a choice for himself or, more importantly, to use just any “word”. On the contrary, one enters into Hesychastic spiritual work with the blessing of a spiritual father, and the word to be used, most often the Name of the Word, is necessarily sacramental in character, bearing an intrinsic, deifying power of its own.
The key to any path of invocation, whether Christian or otherwise, lies in the fact that the Name is the Named. On their own, uninformed by a truly uncreated energy, our own meager efforts to concentrate and be attentive count for little.