Bless Father Hieromonk Ambrose!
Forgive me, but while there are doubtless Orthodox circles where the Rule of St Seraphim to the Theotokos is not practiced, there are sources galore that present it and there are many Russian monastics in Russia (can't speak for North America) that do.
I know EC professors of theology who recite it - each has his own way of praying it just like there are many ways of praying the Rosary. In fact, ways of praying the Rosary can differ according to various Catholic religious Orders (e.g. Dominican, Servite, Carmelite etc.).
In fact, the Orthodox Rule of the Theotokos, in terms of form, is exactly like the western Rosary, except that the Mysteries to be contemplated differ slightly Again, the western Catholic rosary can have any different number of mysteries as well.
My correspondent at Diveyevo in Russia assures me the 150 Hail Mary's, interspersed with an Our Father, is recited DAILY not only by the nuns there, but also by the thousands of pilgrims who go there yearly - and they continue this practice throughout their lives. On feast days, the nuns and pilgrims at Diveyevo actually SING the 150 Hail Mary's as they walk around the monastery three times, prayer ropes and lestovkas in hand.
I just took from my bookshelf a Russian Orthodox prayerbook "Pravoslavniy Molitvoslov" with the approval-recommendation of the "Isdatelskym Sovetom Russkoy Pravoslavnoy Tserkvy" where the Rule of the Theotokos, all 15 desyatky are laid out. Each decima of Hail Mary's is preceded by a Troparion to be recited that is a reflection on the event in the life of our Lord and/or the Mother of God.
The Russian Orthodox encyclopedia, published by the Moscow Patriarchate in 2003, has the same schema of prayer laid out. More to this, St Seraphim taught that the daily recitation of this rule is MORE important to obtain the protection of the Mother of God than any other prayer to her, be it an akathist, canon or what-not. Again, I would trust the encyclopedia and other published sources with the approval of the Russian Orthodox Church above many Orthodox internet chat forums on this or any other subject.
St Seraphim Zvezdinsky prayed this Rule daily and said it was an ancient tradition of the Russian Orthodox Old Rite (in the sixth chapter of Staretz Zechariah: An Early Soviet Saint, it is said that this rule of prayer was revealed to a monk in the Thebaid in the 8th century and that, at one time, "all Christians prayed it daily.").
I've ordered a "Theotokos" lestovka from Diveyevo where they insert a copy of the Orthodox "Hail Mary" into each step for the recitation of the Rule of the Theotokos.
There are of course many Roman Catholics that don't say the Rosary and, frankly, I've never said it using my "imagination" in prayer. I have icons that depict the event/mystery to be contemplated and I use the troparia schema above with clauses inserted into each "Bohoroditse Divo."
No one is forced to use this form of prayer, of course.
I just hotly (and respectfully) contest your assertion that the Rule of the Theotokos doesn't exist (as you couldn't find it in four internet Orthodox forums you examined).
Internet forums provide great services to people. However, they are a very poor teaching resource for the most part. I belonged to one which was manned almost exclusively by converts who would definitely see the St Seraphim's Rule as a "Latinization" and would stay away from it for dear life. They had a definite anti-Western, anti-Catholic axe to grind. That is also a poor way to define what is genuinely Orthodox, in my view.
It's a good thing the Jesus Prayer isn't widely practiced in Roman Catholicisim as such converts might want to distance themselves from that Prayer as well in their desire to move away from their previous ecclesial association.
That simply isn't so, Venerable Father.
If you will go to this link:
http://lestovka-diveevo.ru, there are actual pictures of hundreds of Orthodox Christians there walking the pathway around Diveyevo, reciting the Rule of the Theotokos.
But please don't take the word of this Eastern Catholic (who stopped praying the Rosary and only came back to it under the influence of writings like Staretz Zechariah et alia).
Please feel free to write Diveyevo to ask them as many questions as you like about the Rule of the Theotokos, who practices it, how widespread it is and the like.
They are very communicative and will be more than happy to provide you with all sorts of information.
Reverencing your right hand (into which I desire to place a Theotokos Lestovka), I again implore your blessing,
Alex