Bpbasil:
Why then do I often hear complaints that the Rosary is prayed as a “latinization” among Eastern Catholics and some Orthodox? …
I know that this was not addressed to me so I apologize in advance for that, but I have to agree with
bpbasilphx that “
The Prayer Rule of the Theotokos is a beautiful devotion, but it should NOT be substituted for the public services of the Church”.
One has to understand that in the Greek east, the Offices are properly a communal parish function wherever and whenever it can be arranged. The Divine Offices (LOTH) are the official prayer of the church. Everyone should make an attempt to participate in the full liturgical cycle (if possible), and most especially for Sundays and other holy days. (The
litourgia is also catechetical, which is an important dynamic too.)
[An illustration of how important an issue this is, is that according to the CCEO attendance at Vespers can be considered sufficient to satisfy the Sunday obligation for an Eastern Catholic, although this provision is not made in the Latin code.]
The problem with the rosary is that it is a private devotion that was (somehow) introduced in place of the Divine Office. It was such a commonplace abuse among Eastern Catholics in Europe that one might think that there was a deliberate plan in place (no one seems to know

).
This has wreaked havoc with the traditional spirituality of the people, which some Eastern Catholic priests and deacons are manfully struggling to restore these days.
I think that the modern Latin Catholic **
GENERAL INSTRUCTION OF THE LITURGY OF THE HOURS **expresses this communal concept quite well, it seems that restoring this is seen as desirable in the west as well.
Celebration in Common 20. The liturgy of the hours, like other liturgical services, is not a private matter but belongs to the whole Body of the Church, whose life it both expresses and affects. This liturgy stands out most strikingly as an ecclesial celebration when, through the bishop surrounded by his priests and ministers, the local Church celebrates it. For “in the local Church the one, holy, catholic, and apostolic Church is truly present and at work.” Such a celebration is therefore most earnestly recommended. When, in the absence of the bishop, a chapter of canons or other priests celebrate the liturgy of the hours, they should always respect the true time of day and, as far as possible, the people should take part. The same is to be said of collegiate chapters.
21. Wherever possible, other groups of the faithful should celebrate the liturgy of the hours communally in church. This especially applies to parishes - the cells of the diocese, established under their pastors, taking the place of the bishop; they “represent in some degree the visible Church established throughout the world.”
Code:
22. Hence, when the people are invited to the liturgy of the hours and come together in unity of heart and voice, they show forth the Church in its celebration of the mystery of Christ.
It must be stressed that the rosary in itself is not a problem (although I am pretty sure most Orthodox do not believe it was handed down by the Theotokos to Saint Dominic nor in the fifteen ‘promises’). It is a magnificent chaplet in it’s own right and in some communities it is quite popular.