There are things about Orthodoxy that a Protestant may find wrong-minded and in some cases even heretical. And since for the Protestant reader at first introduction Orthodoxy looks like Catholic, let me say first that there is about as much difference and disagreement between Orthodoxy and Catholicism as there is between Protestantism and Catholicism. The Orthodox have never had a Pope, for one thing. Orthodoxy and Catholicism separated in the 11th century, thereabouts. Then in western Europe Protestantism split off from Catholicism starting, say, in the 1400s.
The next thing I should say is that Orthodoxy is centered on the worship of the triune God – God the Father; God the Son, Jesus Christ; and God the Holy Spirit. You can learn a lot about Orthodox theology through its worship services.
Moving on to he potential sticking-points, here are some Orthodox practices that Protestants often find wrong-headed:
Apostolic Succession: New bishops are raised to position by existing bishops and the Church hierarchy more generally, and this goes back to the time of the apostles. Not in the apostolic succession? Not the Church … unless there is an exception. For this reason, Orthodox are not to take communion with non-Orthodox … unless there is an exception.
Tradition: Orthodoxy is not sola-scriptura. It teaches that, while Church tradition will not contradict scripture, Church tradition, prayer, and a godly life are needed to properly frame and understand what the scriptures mean. It was, after all, Church councils (made up of prayerful, godly men) that determined what should and should not be in the canon of scripture that we call the bible today. The Orthodox include the apocryphal books that the Catholics use, plus two more.
Prayer to the Saints: I say it that way for the shock value. The Protestant view is that only Jesus should be prayed to and so “prayer” to the saints is heresy. But the Orthodox take “the cloud of witnesses” (Hebrews 12:1) to literally exist, and so regards the saints that have passed on as alive. Based on that we ask these our brothers and sisters in Christ to pray for us. It looks like prayer because they are “dead” and we can’t see them. It also looks like prayer because (if you read the lives of the saints you will understand this) the saints are regarded as greatly spiritually empowered through their godly lives. The motive is partly like when people, in the time of the apostles, stood where the shadow of Peter would touch them so they could receive a blessing of healing or to meet some other need. It also seems like prayer because much is said about and to the saints in church services. But in the services, in addition to “prayers” for help, there is much content that teaches and encourages a godly way of life by describing their lives, motives, and deeds.
Veneration of Icons (and relics): This is an interesting one. It is regarded by Protestants as the worship of idols. I like to get the veneration of icons this way: Would you, if you had been a soldier at war in a foreign land and thought that you would not live to return home, possibly kiss the earth when you returned to American soil or to your own home? Would you if under persecution as a Christian have been unable to see a bible for some long time, kiss a bible if by some miracle God provided you one? Icons represent to the Orthodox that great and heavenly land and kingdom we are presently exiled from. Icons all represent born again people and/or angels and when we bow before them and kiss them we are showing our love and respect. It’s also a good time to ask them for a blessing (see “Prayer to the Saints” above). As for the veneration of relics such as the bones of the saints, all Christians respect the remains of the dead, but this is extended in Orthodoxy through its perception of the nature of the saints and through the many stories of miraculous events circulating around their remains (think of 2 Kings 13:21). It also extends to the cross in its various forms and to various objects such as pieces of the actual cross of Christ, wine jars from the wedding in Cana, any of the sites that Jesus visited, the seemless tunic worn by Christ, or a garment worn by a saint – that sort of thing. Then of course there is the Holy Fire.
High Church (Pomp and Circumstance): The services are designed to use all the senses to teach and to induct the participant into the presence of God … chanting, singing, incense, various implements, vestments, icons …
Deification: Here’s a doozy. Does the Orthodox believer become a god?? The Orthodox call deification what the Protestants call sanctification. It is the process of growing up into an increasing experiential union with and expression of God, the Holy Spirit. But the Orthodox take a very high view of man and have noticed that the fact that God took material form says something both about the nature and capacities of man and about materiality more generally. In Western thought, there is a wall between the material and the spiritual realms. In Orthodox thought they are interpenetrating. It will help to understand the Orthodox concept of deification to recognize that it is part of and weaves together with the Orthodox understanding of saints and relics. In fact the whole of the body of Orthodox thought is one thing. The best way to learn it, as is recommended by many Orthodox, is to “come and see.” It is best learned by experience. But for a crash course, particularly on deification and the saints and relics, you might read The Place of Holy Relics in the Orthodox Church.
I don’t know whether I have missed anything…