Note: This is not urging Eastern Catholic and Orthodox Christians to ‘latinize’ (adopt customs from another Church when they didn’t use them before). However, the Rosary remains a part of us who have come from a tradition where it is common: a veritable school of prayer. For those of us, then, here is a version of this practice nicely adapted to the Byzantine tradition, based on a version devised by Ukrainian Catholic priests of the Basilian Order in the 1950s.
 
In the best Eastern tradition it’s a fine ‘fallback’ to make up for a missed part of the Hours when you’re tired or ill.Some background: Besides the obvious parallel of the Rosary to the prayer beads used by Orthodox monks and nuns, there were Russian adaptations of the Rosary in the 1700s and 1800s. St Seraphim of Sarov is said to have had one. The Russian ‘Rule of the Mother of God’ consists of praying 150 ‘Rejoice, O Virgin Mother of God’ prayers (Hail Marys) a day. Some of the mysteries here are the same as the 12 Great Feasts of the Orthodox Church (Annunciation, Nativity, Ascension, Dormition) while others, especially the Sorrowful Mysteries, have a more Latin emphasis.

You might want to compare this Rosary to the Rule of the Mother of God of St Seraphim (Zvezdinsky), an Orthodox bishop killed by the Soviets in 1937.About the Rosary in general: The Rosary is a practical substitute for the Divine Office/Hours, a sort of layman’s breviary well suited for those who cannot read as was true of many in the Middle Ages. The 150 Hail Marys mimic the 150 psalms. St Dominic is said to have introduced it to the Roman Catholic Church.


    ‘Almost any encyclopædia will inform the reader that the use of beads in prayer is far older than Christianity itself and belongs to almost every race which has any highly developed system of prayer.    ‘It will be a surprise to many, no doubt, to know that our common English word bead is derived from the Saxon word bid, to pray, the derivation arising from the fact that our ancestors made common use of perforated pebbles, or beads, upon which to count their prayers. It will be news even to most Catholics to learn that instead of their Rosaries being spoken of as beads because of a resemblance to the common ornament of the name, the ornament takes its name from the Rosary.’

+ Make the sign of the cross.
The Eastern Orthodox have an equivalent system of Jesus Prayers and prostrations one may use at home that corresponds to the Hours prayed in church.
The Revd S.C. Hughson, Order of the Holy Cross (Anglican), wrote in 1918: 
Start at the crucifix:
+ In the Name of the Father and of the Son and of the Holy Spirit.  Amen.
Glory to Thee, our God, glory to Thee. 
Heavenly King, Comforter, Spirit of Truth, who art everywhere and fillest all things, Treasury of blessings and Giver of life, come dwell within us, cleanse us from all our sins and save our souls, O gracious Lord.
From Easter until Ascension leave out Glory to Thee and Heavenly King and instead say three times Christ is risen from the dead, trampling down death by death, and upon those in the tombs bestowing life. From Ascension until Pentecost, simply make the sign of the cross — In the Name of the Father — before continuing as usual with: 
+ Holy God, holy Mighty, holy Immortal, have mercy on us. (Three times)
+ Glory to the Father and to the Son and to the Holy Spirit, both now and ever and unto the ages of ages.  Amen.
Most holy Trinity, have mercy on us. Lord, cleanse us from our sins. O Master, pardon our transgressions. O Holy One, visit us and heal our infirmities for Thy Name’s sake. Lord, have mercy. Lord, have mercy. Lord, have mercy.
+ Glory to the Father and to the Son and to the Holy Spirit, both now and ever and unto the ages of ages.  Amen.
Lord, have mercy. (12 times)
+ Glory to the Father and to the Son and to the Holy Spirit, both now and ever and unto the ages of ages. Amen.
+ O come, let us worship God our King.
+ O come, let us worship and bow down before Christ, our King and our God.
+ O come, let us worship and bow down before Christ Himself, our King and our God.
Then you may say Psalm 50 (51), Have mercy upon me, O God — or another appropriate psalm — and the Nicene Creed, I believe in one God, the Father almighty.
Then at the first small bead on the pendant:
+ Glory to the Father, who hast made us, both now and ever and unto the ages of ages.  Amen.
Rejoice, O Virgin Mother of God, Mary full of grace, the Lord is with thee. Blessed art thou among women and blessed is the fruit of thy womb, for thou hast borne the Saviour of our souls.
Then at the second small bead on the pendant:
+ Glory to the Son, who hast redeemed us, both now and ever and unto the ages of ages. Amen.
Rejoice, O Virgin Mother of God...
Then at the third small bead on the pendant:
+ Glory to the Holy Spirit, who hast enlightened us, both now and ever and unto the ages of ages.  Amen.
Rejoice, O Virgin Mother of God...
+ Glory to the Father and to the Son and to the Holy Spirit, both now and ever and unto the ages of ages.  Amen.
The decades or mysteries are as in the Latin (Dominican) Rosary, with an Our Father, 10 ‘Rejoice O Virgin Mother of God’ (Hail Mary) prayers and the ‘Glory to the Father and to the Son and to the Holy Spirit’. After each decade one may say the Fátima prayer: O my Jesus, forgive us our sins, save us from the fires of hell. Lead all souls to heaven, especially those most in need of Thy mercy.
The mysteries are:JOYFUL: Annunciation, Visitation of Our Lady to St Elisabeth, Nativity of Christ, Presentation of the Child Jesus in the Temple, Finding of the Boy Jesus in the Temple.  SORROWFUL: Agony in the Garden, Scourging at the Pillar, Crowning with Thorns, Jesus Carries the Cross, Crucifixion.
GLORIOUS: Resurrection, Ascension, Descent of the Holy Spirit, Dormition/Assumption and Crowning of Our Lady in Glory in Heaven.
Traditionally you pray the Joyful Mysteries on Mondays and Thursdays, the Sorrowful Mysteries on Tuesdays and Fridays, and the Glorious Mysteries on Wednesdays, Saturdays and Sundays.
In 2002 the Pope added another set of mysteries that may be prayed on Thursdays, changing Saturday’s set to another round of the Joyful ones.
LUMINOUS: Baptism of Christ in the Jordan (Theophany, Богоявление), the Wedding at Cana, Jesus Proclaims the Kingdom, the Transfiguration (Преображение) and the Institution of the Eucharist.
Closing Prayer
JOYFUL MYSTERIES: It is truly proper to glorify thee, who hast borne God, the ever-blessed and immaculate and the Mother of our God! More honorable than the cherubim and beyond compare more glorious than the seraphim art thou, who without corruption gavest birth to God the Word. Thee, truly the Mother of God, we magnify!
SORROWFUL MYSTERIES: We fly to thy patronage, O holy Mother of God. Despise not our prayers in our necessities but deliver us from all danger, O glorious and ever-blessed Virgin.
+ Most holy Mother of God, save us. (Three times with bows)
GLORIOUS MYSTERIES: To thee, the Champion Leader, we thy servants dedicate a feast of victory and thanksgiving as ones rescued out of sufferings, O Mother of God: but as one with invincible might, from all dangers deliver us that we may cry to thee: Rejoice, O Unwedded Bride!
This last prayer is from the Akathist to the Mother of God, an important Marian devotion in the Orthodox Church.