I am always struck by how much Mary is invoked in the Eastern Byzantine liturgy (which is such a contrast to the Roman liturgy, which rarely makes explicit reference to Mary). Even though they do not have the tradition of the Rosary, the presence of Mary in their prayer traditions is so pervasive that she surely gives them great blessings and ardently desires all to be in full communion with the Successor of St. Peter.
Listen to how the all the elements of the Western "Hail Mary" are contained in a beautiful way in the prayers for the Feast of the Dormition:
"Hail, O Woman full of grace, the Lord is with you, the Lord who, because of you, bestows great mercy upon the world" (from Vespers). "Blessed are you among women and blessed is your womb that contained Christ, in whose hands you committed your soul. O pure Virgin, intercede with Christ our God that He may save our souls!" (at the Apostichon).
How can Mary not want them to have the fullness of unity with us?
So why hasn't it happened? We must not be praying hard enough! We are not loving, learning, or sacrificing enough of ourselves. They need to see the face of love, the face of Christ, in the faces of those who adhere to the Catholic Church. The Orthodox peoples of the Middle East, Eastern Europe, and in a special way those mysterious people beloved of God--the Russians--need Peter in order not to lose what they have, and in order to give to the whole world the riches of their faith!