If you mean Russian, Greek or Coptic or Armenian ... some will accept converts, others don't. Each has it own learning schedule, testing, and even catechism and different meanings to Jesus, Mary and the trinity. So it can take a year or more of learning.
Attend Liturgy. Tell the priest you would like to be a coachmen. Learn all you can. The reason why you need to learn so much is because in order to be an Orthodox Christian, you must agree with the teachings of the church, so they want to make sure that you are sure. There is no hurry.
It is the absolutely most exciting time in all my life. After 30 years as a non-den om charismatic, I converted in the spring of 2008. If there is anything I can help you with, e mail me. Check out beta.ac rod.org and go to the "Orthodox Reading Room" for a vast amount of info. Have fun. and the site will not be a beta much longer. It is scheduled to be just ac rod.org in the next week or two. There is also go arch.com
To become orthodox you will need to become an inquirer first. Find out where all the orthodox parishes are around you and visit them each individually. Speak with a priest about your interest. Typically you go to inquirers classes for at least a year and then you are a coachmen for a brief period of time before you are baptized and charismatic. Our classes at my parish are once a week winter Thur spring.
Inquirers classes are essential to properly understand orthodoxy. You may be able to be brought into the church earlier or later depending upon the priest you meet and how ready he feels you are. In fact in a good deal of parishes you will just meet with the priest until he feels you are ready.
I am unaware of any branch of the orthodox church that does not accept converts. But you will find parishes that are more "ethnic" than others so they may have you jump Thur more hopes. Typically here in the US the OCA (http://www.oca.org/) and the Antiochian (http://www.antiochian.org/) are the least ethnic of the various orthodox groups. The Antiochians have the most protestant converts since there was a mass conversion of entire protestant churches over the last couple decades and the Antiochians were the ones to bring them in.
Ethic parishes are NOT bad and they will all accept you if you have a true interest in conversion. My priest is a Jewish convert to protestantism and then to eastern orthodoxy.
Feel free to message me for more specific questions if you like.
Many years!
(and as an aside the Coptic and Aremenian Orthodox Churches are actually not technically "Eastern" orthodox, they are oriental orthodox.)