Dear Paul,
In fact, the Orthodox have the Rosary, but call it the "Rule of Prayer of the Theotokos."
St Seraphim of Sarov and St Seraphim Zvezdinsky, among many others, prayed this Rule which consists of 15 groups of ten Hail Mary's each headed by an Our Father. The ending prayer of each group of ten or decade is: Open to us the doors of mercy, Blessed Theotokos, so that having hoped in Thee, we should not perish, but would be saved by Thee from troubles, for You are the salvation of the Christian people. (Ok, my translation may not be the best . . .).
The points of reflection for each of the 15 decades are:
The birth of the Theotokos
Her entrance into the Temple
Her Annunciation
Visitation to St Elizabeth
Nativity of Christ
Presentation of Christ in the Temple
Flight into Egypt
The finding of Christ in the Temple
The Wedding in Cana
The Crucifixion
The Resurrection
The Ascension
Pentecost
The Dormition
The Coronation of the Theotokos and her Protection
St Seraphim would say that this form of prayer was revealed to a monk in the Thebaid by the Mother of God in the 8th century and it soon spread all over the Church.
In fact, "Rosary" as a term was actually disliked by many Western Catholic saints who preferred to call it the "psalter of the Mother of God."
St Seraphim always insisted his spiritual children pray this Rule daily and at the monastery of Diveyevo in Russia, the nuns and all visitors walk around the ditch surrounding the monastery, reciting the Hail Mary's. On feast days, the nuns sing the Hail Mary's.
So while Orthodox may be against the Rosary, they are all for the Rule of Prayer of the Theotokos!
Alex