Yes, Centering Prayer is a rip-off of TM. But there are precursors remarkably like TM in the earlier Church, especially among the Quietists. The 17th-century Miguel de Molinos taught us to turn our attention inward without imagination and adopt a passive attitude to whatever arises in your mind (soul). One aspect that reminds me of TM is that Molinos assures us there is no reason to worry when thoughts are distracted, because it is not necessary to think of God (ie, to have one's attention turned inwards) during the whole time of the prayer. The prayer still continues provided that one has been sincere in starting the session - it's up to God what happens during the allotted time. Sounds similar to the TM idea one should favour the mantra but be relaxed about allowing other thoughts to arise. He was found guilty of inappropriate sexual contact with his female disciples (sound familiar?). He admitted the charge but claimed they were "purifying acts caused by the devil". Ah yes! We've all had those. He was condemned for heresy and died in prison.
The 14th-century English classic The Cloud of Unknowing also has parallels with TM and advises repeating a one- or two-syllable word. So a passive attitude and a "mantra" - all that's missing is an initiation fee.