I know of at least one Greek Catholic who prays (or at least ATTEMPTS) to pray the rosary: me. As distracted as I so often am all I can do is attempt prayer of any kind.

I try, however, to "byzantinise" it by commencing with the "usual beginning" prayers and by interspersing, between decades, troparia and other short Byzantine liturgical texts I have memorised.

Usually at the end I conclude by praying one of the "dostoyno jest'" texts we sing after the consecration of the Eucharist in our Liturgy, instead of the beautiful "Hail, holy Queen".

I don't know if this practice does me any good but it don't hurt nothin', either.

There are undoubtedly many Eastern Catholics who pray the rosary, but no, it is not a part of Eastern tradition. Indeed, I was appalled to learn, when visiting the website of a Ukrainian church in Canada, that they actually have a scheduled rosary in the parish. I'm sure they are not alone.

For all that, on a corporate basis at least, praying the rosary is explicitly discouraged by Rome, insofar as it is yet one more remnant of the Latinised past. And that is what you are testifying to (above) when you mention all of those Eastern Churches that pray it.

If an Eastern Christian wants to pray it as an individual, there is no real problem. It may well be good for them. But no, the rosary is NOT something that finds a place in Eastern Catholic tradition.

"You'll freeze to death!"
from an interview with Maria Bashynska
When I was in Siberia, once they told us this story : there were three sisters there in the prison, and they were bitterly persecuted because they prayed. And the commandant who was in charge there, well, he was something awful. He wanted them to stop praying, and he wanted them to renounce their faith. Let me tell you, it wasn't going to happen! So, he separated them and put each one in a different room for a few days. But they didn't stop. Then he said to them: "It'll be 60 degrees below zero, you'll go out in the cold and you'll freeze to death!"
One day the temperature dropped to 60 below. The soldiers were bundled up in heavy leather coats and boots and winter hats. They brought the sisters out into the barracks yard, barefoot and in shirt-sleeves and ordered all the prisoners out to watch the nuns freeze to death. The prisoners came out to watch. There were dogs near the soldiers, three dogs.
The commandant swore that in half an hour they would freeze to death. The sisters knelt down to pray the rosary. Half an hour passed. They didn't freeze. They continued kneeling and all three prayed loudly. Then he set the dogs on them to devour them. The dogs rushed up to them, circling around the sisters. But then they lay down in the snow and wagged their tails. They didn't do anything bad to the sisters; they cuddled up to the sisters.
Everyone could see that the dogs weren't going to do anything to them, and the prisoners started to shout and sing "Praise be to God!" When the prisoners began to sing, the guards told the sisters: "Go back to your cell." The sisters went back, and they never had any more trouble, and from that day on they prayed with the people as much as they wanted.
Maria Bashynska (Sr. Markiya, SSMI) was born in 1917 in the village of Hutysko, Zhovka District, Lviv Region.
(File P-1-1-215)

Source: http://www.ugcc.org.ua/33.0.html?&L=2