Orthodox wedding service is not only ancient, but also beautiful and filled with rich symbolism. Below you will find explanation of why we put the rings on our right hand during the betrothal (engagement).
Orthodox wedding is composed of two ceremonies: the betrothal and the crowning, which today (most of the time) are celebrated as one service. The betrothal service, according to Father John Meyendorff, is the “…new form of marriage contract… It was originally a civil ceremony.” 
The Betrothal service is the exchange of rings
Interestingly, it is in the betrothal service (i.e, the official engagement) that the rings are exchanged! Though this is usually the culmination of the wedding service in western Christianity, the Orthodox Church builds up to a much more important moment: the crowning!
But the mystery of the Orthodox betrothal service doesn’t stop there! Not only are the rings exchanged at the betrothal, the rings are put on the right hand! And a priest, not your spouse, puts them there!
Wait a minute! Aren’t the rings supposed to go on the left hand?
Why do the rings go on the right hand? What does all this mean? What do the rings signify?
First let’s look at the meaning of the rings.
Rings as faithfulness
We are typically taught, by secular society, that the rings represent our faithfulness to our spouse, or our trust in our spouse, or a pledge of our love for our new spouse.
However, none of the prayers used in the wedding service suggest this meaning!
The rings represent the faithfulness of God
The references from the wedding service come from the Bible, and they suggest that the ring is a sign of God’s faithfulness! Here are the biblical references, along with what the prayers say,
The prayers starts this way:
Therefore, O Lord our God, having sent forth Your truth upon Your inheritance, and Your promise to Your servants, our fathers, who were Your elect from generation to generation: Look upon Your servant, [groom'sname], and Your handmaid, [bride'sname], and establish and make firm their engagement in faith and in oneness of mind, in truth and in love. For You, O Lord, have declared that a pledge should be given and confirmed in all things.
Joseph gets a ring
St Joseph, the all-comely (Old Testament)
Then, it moves on to mention Joseph, who was sold into slavery in Egypt, but then rose to great power in Egypt: “By a ring power was given to Joseph in Egypt…”
And Pharaoh said to Joseph, “See, I have set you over all the land of Egypt.” Removing his signet ring from his hand, Pharaoh put it on Joseph’s hand... Thus he set him over all the land of Egypt. (Genesis 41:41, 43)
Daniel gets a ring
Holy Prophet Daniel
Then the prayer moves on to mention Daniel, who was a prophet in exile in Babylon. When the Babylonian king forbade Daniel to pray, Daniel remained faithful to God and prayed anyway; for this reason the Babylonian king sealed Daniel in the lion’s den with a ring: “…by a ring was Daniel glorified in the land of Babylon…”
A stone was brought and laid on the mouth of the den, and the king sealed it with his own signet ring and with the signet ring of his lords, so that nothing might be changed concerning Daniel. (Daniel 6:17)
Tamar gets a ring
Then the prayer mentions Tamar, who was denied her legal rights to a Levirate marriage. She tricked Judah into sleeping with her by pretending to be a prostitute. However, she asked for his ring, so when Judah found out she was pregnant, she could prove it was his and that she was within her legal rights: “…by a ring was the truth of Tamar revealed…”
He said, “What pledge shall I give you?” She replied, “Your signet ring and your cord, and the staff that is in your hand.” So he gave them to her, and went in to her, and she conceived by him. (Genesis 38:18)
The prodigal son gets a ring
The return of the prodigal son
Finally, the prayer mentions the Prodigal Son in the New Testament, who had disrespected his father, ran away from home, but then returned in repentance: "…by a ring our Heavenly Father showed His compassion upon His Prodigal Son, for He said: 'Put a ring on his finger and get the fatted calf and kill it, and let us eat and celebrate'."
But the father said to his slaves, ‘Quickly, bring out a robe—the best one—and put it on him; put a ring on his finger and sandals on his feet. (Luke 15:22)
In other words, the rings symbolize God’s word that GOD will seal our marriage in “…faith, in oneness of mind, in truth, and in love.”
The ring is a God's pledge to us
It short, the rings symbolize God’s pledge to us, His children.
God promises to remain with us, whether we are sold into slavery (like Joseph), thrown into the lion’s den (like Daniel), denied our legal rights (like Tamar), or when we return to him in repentance even if we formerly rejected Him (like the prodigal son). This is what the rings signify.
The right hand of power
But why the right hand? The prayer for the ring also mentions Moses:
God's right hand giving the Ten Commandments to Moses
Your own right hand, O Lord, armed Moses in the Red Sea, by Your true word the heavens were established, and the foundations of the earth were made firm, and the right hands of Your servants will also be blessed by Your mighty word and by Your uplifted arm.
Moses’ right hand was, in fact, God’s hand, which saved the Hebrews through the waters of the Red Sea. God’s right hand “makes firm” the foundations of the earth. Thus, by putting the rings on our right hand, rather than our left, we are reminded that everything we do is with the help of God, who watches over us and protects us!
P.S. the rings are about God's faithfulness to us
The prayer for the rings ends in this way:
“Therefore, O Master, Yourself bless now this putting-on of rings with Your heavenly blessing, and let Your angel go before them all the days of their life. ForYou are He Who blesses and sanctifies all things…”
The rings, in the Orthodox tradition, mean much, much more than simply, “I love you.”
They symbolize God’s word to remain faithful to us, and seal our marriages in faith, in oneness of mind, in truth and in love.
They also symbolize that God walks with us, protects us, and supports us. Our marriages unite us to our spouse, but they also proclaim the gospel message, and remind us of the love God has for us, His children!