An Eastern Orthodox rosary (the beads not the prayer) is known as a "Chotki". This Oriental form of rosary is known in the Hellenic Greek Church as Chaplet or String of Knots of Beads(komboskini) in the Russian Church as Vervitza (string), chotki (chaplet), or liestovka (ladder), and the the Rumanian Church as matanie (reverence). Our everyday name of "beads" for it is simply the Old Saxon word "bede" (a prayer) which has been transferred to the instrument used in reciting the prayer, while the word rosary is an equally modern term.
The "Jesus Prayer" Rosary prayed by the Eastern Orthodox is a string of 33, 100, or 300 beads on a string or knots made of wool; they are not divided into decades. (I have a 33 knot wool chotki bracelet which travels nicely on the wrist.)
On each bead or knot is prayed the following "glorious" mantra: "Lord Jesus Christ, Son of the Living God, have mercy on me, a sinner." The modern Chotki, as it is known, calls for a slightly different mantra: "Lord Jesus Christ, Son of God, have mercy on me, a sinner." Both this mantra and the Glorious mantra above are derived from the Gospel story of the Pharisee and the tax collector.
Two men went up to the temple to pray, one a Pharisee and the other a tax collector. The Pharisee stood and prayed thus with himself, 'God, I thank Thee that I am not like other men--extortionists, unjust, adulterers, or even as this tax collector. I fast twice a week; I give tithes of all that I possess'. But the tax collector, standing afar off, would not so much as raise his eyes to heaven, but beat his breast, saying, 'God, be merciful to me a sinner!' I tell you, this man went down to his house justified rather than the other; for everyone who exalts himself will be humbled, and he who humbles himself will be exalted. (Luke 18.10-14)
and listen to St. Andrew of Crete: 'Boastful am I, and hard of heart, all in vain and for nothing. Condemn me not with the Pharisee, but rather grant unto me the humility of the Publican, O only merciful and just Judge, and number me with him' (Great Canon, Ode 4).
(The Hail Mary, of course, comes from the Angelic Salutation coupled with a later additional prayer.)
From Bishop Kallistos Ware:
QUOTE There is one type of private prayer, widely used in the west since the time of the Counter-Reformation, which has never been a feature of Orthodox spirituality - the formal 'Meditation,' made according to a 'Method' - the Ignatian, the Sulpician, the Salesian, or some other. Orthodox are encouraged to read the Bible or the Fathers slowly and thoughtfully; but such an exercise, while regarded as altogether excellent, is not considered to constitute prayer, nor has it been systematized and reduced to a 'Method.' Each is urged to read in the way that he finds most helpful.
But while Orthodox do not practise discursive Meditation, there is another type of personal prayer which has for many centuries played an extraordinarily important part in the life of Orthodoxy - the Jesus Prayer:
"Lord Jesus Christ, Son of God, have mercy on me a sinner."
Since it is sometimes said that Orthodox do not pay sufficient attention to the person of the Incarnate Christ, it is worth pointing out that this - surely the most classic of all Orthodox prayers - is essentially a Christo-centric prayer, a prayer addressed to and concentrated upon the Lord Jesus. Those brought up in the tradition of the Jesus Prayer are never allowed for one moment to forget the Incarnate Christ.
As a help in reciting this prayer many Orthodox use a rosary, differing somewhat in structure from the western rosary; an Orthodox rosary is often made of wool, so that unlike a string of beads it makes no noise.
The Jesus Prayer is a prayer of marvelous versatility. It is a prayer for beginners, but equally a prayer that leads to the deepest mysteries of the contemplative life. It can be used by anyone, at any time, in any place: standing in queues, walking, traveling on buses or trains; when at work; when unable to sleep at night; at times of special anxiety when it is impossible to concentrate upon other kinds of prayer. But while of course every Christian can use the Prayer at odd moments in this way, it is a different matter to recite it more or less continually and to use the physical exercises which have become associated with it.
Orthodox spiritual writers insist that those who use the Jesus Prayer systematically should, if possible, place themselves under the guidance of an experienced director and do nothing on their own initiative.
For some there comes a time when the Jesus Prayer 'enters into the heart,' so that it is no longer recited by a deliberate effort, but recites itself spontaneously, continuing even when a man talks or writes, present in his dreams, waking him up in the morning.
In the words of Saint Isaac the Syrian: 'When the Spirit takes its dwelling-place in a man he does not cease to pray, because the Spirit will constantly pray in him. Then, neither when he sleeps, nor when he is awake, will prayer be cut off from his soul; but when he eats and when he drinks, when he lies down or when he does any work, even when he is immersed in sleep, the perfumes of prayer will breathe in his heart spontaneously' (Mystic Treatises, edited by Wensinck, p. 174).
Orthodox believe that the power of God is present in the Name of Jesus, so that the invocation of this Divine Name acts 'as an effective sign of God's action, as a sort of sacrament' (Un Moine de l’Église d’Orient, La Priére de Jésus, Chevetogne, 1952, p. 87). 'The Name of Jesus, present in the human heart, communicates to it the power of deification ... Shining through the heart, the light of the Name of Jesus illuminates all the universe' (S. Bulgakov, The Orthodox Church, pp. 170-171). UNQUOTE