Here are two others that come to mind:

Valley of Our Lady Monastery: http://www.nunocist.org/
"Seven times a day we gather in choir to pray the Divine Office, a prayer composed of psalmody, scripture reading and hymns. Our Office is sung in Latin with Cistercian chant."

St. Michael's Abbey, Silverton, CA: http://abbeynews.com/Abbey-Home.html
"At St. Michael’s the daily liturgy occupies a large portion of each day. All 7 canonical hours of the Office are sung in their entirety each day, and there is a daily Holy Hour and Benediction of the Blessed Sacrament every day. On a normal day, priests and seminarians spend 3 hours in choir. This time increases on feasts and solemnities." They sing Norbertine chant.

Sam Schmitt

Traditional monasteries are the only kind we have, well, with the exception of New Skete (yes, the monastery that writes the dog training books), but I believe the New Skete monks are all converts. I find it interesting, however, that they organized as a skete, given their tendency to be, uhm, odd.

On either end of the spectrum, you have hermits -- who live alone -- and a cenobitic community, which is what most people think of when they think of monasteries. A skete is halfway between the two. It's a community where the inhabitants gather for liturgical prayer or meals, but otherwise, spend time alone like hermits.

Other than those three (hermitage, skete, cenobitic community) there are no variations in Eastern monasticism. No orders, or anything like that.

From the Petersham Monastery website:

Our celebration of Mass is according to the Missal of Vatican II, not the Tridentine. Only on Thursdays do we have Mass prayers completely in Latin. On other days the prayers are in English while the chants alone are in Latin, i.e., the entrance hymn, the Kyrie (Greek), the Gradual (which for us replaces the Responsorial Psalm) and/or the Alleluia, the Offertory hymn, the Sanctus (Holy Holy Holy), the Agnus Dei (Lamb of God), and the Mystery of Faith. On Sundays and Solemnities, the Gloria (Glory to God) and the Creed are also chanted. All these chants are taken from the approved books produced since the Second Vatican Council by the Abbey of Solesmes in France, the main proponent of Gregorian Chant both before and after the Council. Most date to the tenth to fifteenth centuries, while some hymns of the Office date back to as early as the fourth.

Link to Ryan M.