This is also something I have been seeking. A friend of mine ended up converting to the Orthodox Church, but the Parish he joined seemed very “cultish” after I attended Diving Liturgy with him one Sunday, and so I turned down his offer to speak to his Priest about becoming a Spiritual Director. There is a Melkite Church in my city, would that be a place to start looking for a spiritual director? I’ve heard that some Orthodox Priests won’t even consider teaching The Jesus Prayer to a Catholic unless they convert.
Priests can be a good place to start when one is seeking a spiritual father, but it must be noted that not all priests are qualified to be spiritual fathers simply because they are ordained. I have heard many sad stories about people turning to their pastors for spiritual direction, only to be led in harmful directions by their priest, who thought ordination automatically made him a qualified spiritual father. Monks can be the same way, and I’ve heard that they are often worse, ordering their directees not to receive Holy Communion for no real reason. It is important to remember that ordination and/or the monastic habit do not automatically make one a qualified spiritual father.
What you want to look for is a person, man or woman, ordained, monastic or lay, who is experienced in prayer and spiritual warfare. This person must be willing to take your spiritual journey with you and not set himself over you.
I would ask around the Melkite parish in your city. Perhaps the pastor could direct you, or at least point you towards one of the deacons or lay people who would make a suitable spiritual father or mother.
Send me a PM and let me know what city you live in. Perhaps there is a monastery or some such near you where you could go to the monks.
Finally, if you cannot find a spiritual father or mother, then read, read, read. I would start with “The Way of the Pilgrim” (although I do advise a little caution with that book), “The Art of Prayer: An Orthodox Anthology”, and/or “On the Prayer of Jesus”. All of these books are wonderful introductions to the practice of the Jesus Prayer. Probably the most complete of the three would be “The Art of Prayer”. It’s a handy little volume that can be picked up at any time and opened to any page and you are guaranteed a juicy morsel from the Fathers that will give you plenty to pray and meditate over for the rest of the day. It’s selected quotes from the Fathers on all aspects of the spiritual life with a particular focus on the Jesus Prayer. Most of the quotes are very short and can be easily read even when you are on a break at work. I cannot recommend this book highly enough.
There is also a new film with accompanying book that I would recommend. I do not remember the title, but I believe it is something like “The Mysteries of the Jesus Prayer”. Perhaps one of our more knowledgeable forum members knows the title. If you could find this I think you’d be in good shape.

My local Borders has the book, so perhaps yours does as well.