By way of clarification, Orthodox monks can be priests and bishops.
In fact, they do not cease to be monks when they reach these other offices in the church. Their lay clothing is, in fact, monastic habit.
In the east the monastic houses can be like a parish in relationship to the bishop, really under his authority. The bishop will probably appoint an Archimandrite to oversee them for him, but they are not tied together in any other way (like financially or organizationally).
Previously, monastics were subject to the local bishop in the west as well, or they established themselves in places no hierarchy existed yet. Sometimes during the initial missionary period of Europe, in fact, the local bishop actually was the Abbot of the local monastery.The reason religious orders are not known in the east is that they were developed in the west after the schism. It was part of a general trend subsequent to the Gregorian reformation.
The Benedictines still exhibit many traits of that earlier age. They tend to be very independent from the local bishop but also from each other, their relationship to other houses is cooperative. The Cluniacs and Cistercians, on the other hand, originated like any other Benedictine house, but maintained tight control of daughter houses, thus forming a prototypical order.
As the Papacy began to evolve it started to authorize other types of organizations which could function across diocesan borders. The new orders began to see the bishop of Rome as their primary Patron. In the beginning some local bishops disliked this trend, but they were ultimately replaced by a new breed of bishop, many of whom got their own clerical starts in the orders. This eventually helped centralize control of the church.