Yes, they do, that’s why they are part of the Catholic Church – otherwise they would probably be Orthodox.

“Eastern Rite” is a bit of a misnomer because there are some 20+ non-Latin rites of the Catholic Church, all considered the Eastern rites.

While doctrine is the same all over the Catholic world, the Rites used by which the Sacraments are administered developed differently in the Early Church. In part because of the languages and customs of the areas that the Church was first established by the Apostles, and Disciples in the Early Church. Since these rites are as ancient as the rites used in Rome, they are of equal value, as a treasure of the Church. The reality is any approved rite of the Church, and all the sacraments that are administered in a valid and licit manner are means of grace because our Lord Himself established them.

You may want to look at this webpage about Eastern Catholics.

http://www.melkite.org/eastern.htm

There are major groupings, For example those who use the Divine Liturgy of St. John Chrisostom, include the Melekite, Greek, Russian, Ukranian, (Ruthenian) and a couple of other Sui Juris Churches. They are all in communion with the Pope, and hold the same Faith as all Catholics do, but the family of Rites they belong to is different.

Most Roman, or Latin Rite Catholics do not know that the Latin Rite is not, and has not been a single rite. There still exists today the Ambrosian Rite, (in Milan) and the Mozarabic Rite, (in Toledo Spain) Before the Reformation there were other rites, like the Gallican in France and Germany, as well as two uses in England the Sarum and York. Some Latin Rite Religious Orders had different Rites for the Mass, and other Sacraments prior to Vatican II, a few places still use these rites from time to time, fewer have permission to use the old rites all the time. They included a different rite than the customary Roman Rite used by the Dominicans, Carmelites, Norbertine, and a few other orders.

The Catholic Church has held that Unity is absolutly needed in essentials, (Faith and Morals) but that there is room for diversity among the rituals of the Church.

We also have to remember that as the Church spread to the ends of the world, the Aposltes founded Churches where different languages were spoken. The first Liturgies, (Masses) were offered in the local language. In the City of Rome, and all of the Roman Empite, Greek, not Latin was the language that most people spoke. The New Testament and a few of the books of the Old Testament were written in Greek. The Jews outside of Israel read the Septuagent, (abbreviated LXX) a Greek translation of the Old Testament. In the first few centuries the Churches in Rome used Greek, it was in Northern Africa where Latin was first used among Catholics.