Orthodox worship and devotion is (or strives to be) dispassionate, with stillness, dignity, gravitas and humility, not grand displays of emotion and histrionics. This can be best observed in the traditional chant styles, and in iconography. There simply is no place for "charismatic" styles of worship in Orthodoxy. It would be as incongruous as a priest wearing Day-Glo.
But the "Jesus Prayer" IMHO has a very similar feel to the "private prayer language" of the Pentecostals - i.e. speaking in tongues but in private prayer. What I mean is that it has the same intimacy, not that it's the same in style.
OK, they are both intimate, and the style is different. But so is the substance. According to St Basil, orthodox don't think of the prophecies in the OT as being written in such a way as a "medium" where the person is controlled like a robot. I assume that extends to prayer life. The Jesus prayer is not the same event like being overcome and going into a trance where God's spirit supposedly or in fact speaks through the person like tongues of fire.
Also, I think the Pentecostal thing is probably self-induced, where the person puts themself in a certain mindset and allows their mind to express their subconscious. Such blind expressions can be beautiful, and I suppose holy, but I think there's the possibility that without any discernment or thought, where YOU decide when the holy spirit is going to come onto you... this might not always be very healthy for mind or soul, since demons have appeared as angels even during people's prayer time.
The Pentecost of the apostles had a definite utilitarian purpose of spreading the message and was unexpected. The charismatic movement makes it as a form of singing or praising God and expect it. I am also skeptical of any prophecies they have like Nostradamus or Marian apparitions, whether or not it is in a prayer session, because we must test the spirits.