The problem is when people pray it in church before Liturgy, as a replacement for the service of Orthros (Matins & Lauds) which should be done publicly with the priest (who has a much bigger role in the East than he does in the West with the Hours, since the Hours include blessings and litanies and other things we like

). The biggest destructive effect of Latinizations is that they water down the intensity and fullness of liturgical life.
Most Eastern Catholics, myself included, do pray the Rosary as a strictly
private devotion. I’m lazy and I usually pray the five-mystery abridgement of the Dominican Rosary.

We do have our own 15-mystery Rosary, the “Rule of the Theotokos” or “Theotokos Lestovka”, which was generally said only by the hardcore Old Believers in Russia and the monks on Mount Athos, who probably got it from the Desert Fathers in Egypt (making it as old as monasticism - back to the 4th century). I do not know whether the Greek version has “mysteries”; the version of the Theotokos Lestovka prayed by the Old Believers, or at least by St. Seraphim Svezdinsky, has 15 mysteries slightly different from the Dominican mysteries: the Nativity of Mary, the Entrance of Mary into the Temple, the Annunciation, the Visitation, the Nativity of Christ, the Meeting (Presentation) in the Temple, the Flight into Egypt, the Finding in the Temple, the Wedding at Cana in Galilee, the Sorrow of Mary standing underneath the Foot of the Cross, the Resurrection, the Ascension, the Dormition, and the Protection of the Mother of God.
The beginning and ending prayers of the Byzantine Rosary are different than in the Latin Rosaries (Dominican or Franciscan), and can be found here:
therosemarytree.blogspot.com/2009/01/rule-of-theotokos-byzantine-rosary.html
A much clearer presentation was found on the website of St. Anne’s Byzantine Catholic Church, which I saved to my computer as a Word document, but the website appears to be gone. It was linked on another thread (by a Roman Catholic, I think), in this post:
forums.catholic-questions.org/showthread.php?t=542548