I have mixed feelings regarding the territorial zeal some fellow Orthodox have regarding this matter. The Jesus Prayer has a unique role in Eastern Christian teaching, but it's plaintive intent is not unique to us. I suspect for as many saints and teachers who argue for its exclusivity, you would find others less inclined.
While we and the westerners differ on our views in some important manners, there is a commonality among us and the Romans which both of us, east and west, tend to look past.
For example, the Greek Metropolis of Atlanta offers these prayers of repentance:
"Prayers for Repentance - O God, my good and loving Lord, I acknowledge all the sins which I have committed every day in my life, whether in thought, word or deed. I ask for forgiveness from the depths of my heart for offending You and others and repent of my old ways. Help me by Your grace to change, to sin no more and to walk in the way of righteousness and to praise and glorify Your Name, Father, Son and Holy Spirit. Amen.
O Lord my God, I confess that I have sinned against You in thought, word and deed. I have also omitted to do what Your holy law requires of me. But now with repentance and contrition I turn again to Your love and mercy. I entreat You to forgive me all my transgressions and to cleanse me from all my sins. Lord, fill my heart with the light of Your truth. Strengthen my will by Your grace. Teach me both to desire and to do only what pleases You. Amen. " http://www.orthodoxprayer.org/OtherPrayers.html
Roman Catholics approach reconciliation/confession with this simple prayer:
"O my God, I am heartily sorry for having offended You and I detest all my sins, because I dread the loss of heaven and the pains of hell, but most of all because they offend you, my God, who are all good and deserving of all my love. I firmly resolve, with the help of your grace, to confess my sins, to do penance and to amend my life." http://www.catholic.org/prayers/prayer.php?p=208
In each example, the penitent approaches the mystery from basically the same starting point, that is an open acknowledgement of sinfulness and a sincere desire to lead a life in accord with God's will.
I am simplifying of course, but each acknowledges God, seeks His forgiveness and pledges to do better.
These sentiments are not at odds and are reflective of the Jesus Prayer, "Jesus Christ, have mercy on me a sinner."
In the end, the answer is not a simple one, but rests with the spiritual intent of the individual. After all, the misuse of the prayer may be be dangerous to the spiritual health of an Orthodox Christian as well.