The Trisagion rosary
The Order of the Most Holy Trinity for the Redemption of Captives was founded in France in 1198. Their Rule originally required that one-third of their total income be devoted to purchasing the freedom of Christian captives who had fallen into the hands of slavers or pirates. (More information here and here.)
From an early date, the Trinitarians have used a form of prayer based on the Trisagion (sometimes Trisagium or Triagion, from the Greek “three” + ”holy”). This is a Byzantine prayer in praise of the Holy Trinity: its simplest form is “Holy God, Holy Strong One, Holy Immortal One, have mercy on us.”
The Trisagion rosary (usually called a chaplet) has three groups of nine beads. In reciting the chaplet, each group is preceded by the Trisagion and the Pater Noster. A special prayer is said on each of the nine beads: “To you be praise, glory, and thanksgiving for ever, blessed Trinity. Holy, Holy, Holy, Lord God of power and might; heaven and earth are full of your glory.” Each group of nine prayers is followed by a Gloria Patri ("Glory be to the Father, and the Son, and the Holy Spirit..."), and the whole ends with a closing prayer.
As with other rosaries that are special to a particular religious Order, its history is rather cloudy. The first question is how long the Trinitarians have used the Trisagion and its associated prayers. The prayers themselves are quite old, and may well have come to the Trinitarians from Byzantium through their connections in the Middle East. The Trisagion itself can be traced at least as far back as the Council of Chalcedon(451 AD) and perhaps further. The use of these particular prayers by the Trinitarians may very well date back to the beginnings of the Order.
A separate question is when beads began to be used to count these prayers. Reciting a certain number of prayers does not necessarily imply the presence of beads -- prayers can be counted on one's fingers, by moving a peg from one hole to another, and so forth. One possibility, and the one I'd favor as the most likely time for beads to be introduced, might be the 14th or 15th century -- when other rosaries became popular, suggesting the concept of using beads as counters. Another possibility is the 19th or early 20th century, when rosaries of all kinds became an indispensable center of prayer life for many Catholics.
It's clear that the Trisagion rosary is not closely related to any of the common Western European forms. But while the prayers may have come from Byzantium, I am inclined to think that the Trisagion rosary in its current form is not very closely related to Eastern Christian rosaries either.
The Eastern rosary does not have the same connection to the 150 psalms as its Western cousin, but rather looks to the injunction of Jesus to “pray without ceasing.” The prayer used for the Eastern rosary is most often the “Jesus prayer.” It may be said in longer and shorter forms, a common short form being, “Lord Jesus Christ, Son of God, have mercy on me a sinner.” The number of prayers is most often 99 or 100 (sometimes 101 or 103).
The physical form of the Eastern rosary (Orthodox and Byzantine traditions) seems more often to be knots in a string, rather than beads. While some Eastern rosaries today are made with beads, others are still knotted, sometimes using special knots with symbolic significance. In the East, the rosary is also more of a monastic practice; it doesn't seem to have experienced any great surge in popularity among lay people (unlike the Western forms).
The Trisagion chaplet is unique, and unlike both the traditional Eastern and Western rosaries in almost every respect. Today it is rather uncommon to see a set of Trisagion beads, but they do turn up on eBay and are featured by some artisan rosary makers.
Or, of course, anyone can make their own ;)
ETA: I've been asked for a complete set of the Trisagion prayers used with these beads. The most complete version I've found is here.