What role does hymnody play in non-Western churches?

DISCLAIMER: I am from the USA, but I happen to live in a part of the country that has over 20 Eastern Orthodox churches in a close radius, representing numerous nationalities and distinct ethnic expressions of the faith (for instance, we have multiple Romanian, Serbian and Greek Orthodox churches, and we also have Macedonian, Serbian, Ethiopian, Coptic (Egyptian), and American Orthodox churches).

Eastern Orthodox typically have chanting and sung liturgy, and also choirs - usually a capella but sometimes including simple instruments or a "church bell." You must understand that EVERYTHING centers on the Divine Liturgy in Orthodox worship. It is "the work of the people." The traditional expression concerning Orthodox liturgy is "Always everyone, always together." The OCA website says it well:

"The Divine Liturgy is the common action of Orthodox Christians officially gathered to constitute the Orthodox Church. It is the action of the Church assembled by God in order to be together in one community to worship, to pray, to sing, to hear God’s Word, to be instructed in God’s commandments, to offer itself with thanksgiving in Christ to God the Father, and to have the living experience of God’s eternal kingdom through communion with the same Christ Who is present in his people by the Holy Spirit....

Because the Divine Liturgy exists for no other reason than to be the official all-inclusive act of prayer, worship, teaching, and communion of the entire Church in heaven and on earth, it may not be considered merely as one devotion among many, not even the highest or the greatest. The Divine Liturgy is not an act of personal piety. It is not a prayer service. It is not merely one of the sacraments. The Divine Liturgy is the one common sacrament of the very being of the Church Itself. It is the one sacramental manifestation of the essence of the Church as the Community of God in heaven and on earth. It is the one unique sacramental revelation of the Church as the mystical Body and Bride of Christ."

It is important to understand this, because they do not have a "disconnected" view of worship. In other words, they do not view worship as a conglomeration of singing, reading, etc. It is all one Divine Liturgy. Worship is a participatory experience for all Orthodox Christians gathered. Orthodox tend to view Protestant worship as being a show; something that professional worshippers perform as a service for the people. This runs counter to Orthodox theology, where Divine Liturgy is the common work of the people "on behalf of all and for all." You must also understand that the building carries much significance (oblation and altar tables, icon screen, etc.) and this all ties into Orthodox theology. It seems that Protestant worship is disconnected from its theology. It is acceptable to sing songs with little theological content and preach on any chosen topic. Orthodox follow the Church calendar which prescribes what readings from scripture will occur and many aspects of liturgy will tie in to this "theme" or focus for each gathering. Everything in the service is united, and the Church calendar is designed to teach the entire Christian Orthodox faith to participants over time and then reinforce it. It globally unites Orthodox believers because they can go to a different Orthodox church across the world and they will pick up where they left off. I know that in many Protestant churches, even if you go to the same "denomination" down the street it may be a different message and scripture verse being taught. In Orthodox Christianity this is not so - thus the churches are truly united in belief and practice.

"The living experience of the Christian sacramental and liturgical life is a primary source of Christian doctrine. In the liturgy of the Church, the Bible and the Holy Tradition come alive and are given to the living experience of the Christian people. Thus, through prayer and sacramental worship men are “taught by God” as it was predicted for the messianic age (Jn 6:45).

In addition to the living experience of the liturgy, the texts of the services and sacraments provide a written source of doctrine in that they may be studied and contemplated by one who desires an understanding of Christian teachings. According to the common opinion of the Orthodox Church, the sacramental and liturgical texts—the hymns, blessings, prayers, symbols, and rituals—contain no formal errors or deformations of the Christian faith and can be trusted absolutely to reveal the genuine doctrine of the Orthodox Church." (OCA)

Concerning style, here is a source that says it well:

"Eastern Christianity (the Eastern Orthodox, Oriental Orthodox and Eastern Catholic churches) have a very rich and ancient hymnographical tradition.

Eastern chant is almost always a cappella, and instrumental accompaniment is rare. The central form of chant in the Eastern Orthodoxy is Byzantine Chant, which is used to chant all forms of liturgical worship. Exceptions include the Coptic Orthodox tradition which makes use of the sistrum, and the Ethiopian Orthodox Tewahedo Church, which also uses drums, cymbals and other instruments on certain occasions."

The Orthodox Church has been reevaluating the role of music in worship as of late. One source says,

"Russian Orthodox church music has its particularly unique development. Byzantine music remains basically monophonic (single-line unison singing). But part-singing appeared in Lvov and began to spread in Southern Russia and the Ukraine as early as the 15th Century. From this we can trace early experiments with harmonization, and in the 17th Century the influence of the Kievan schools of harmony on Moscow. Choirs of sorts began to be schooled in the Imperial Court, although they sang in small groups and were made up of male singers only.

It was Peter the Great in the 18th Century who gave rise to the Imperial Chapel Choir. The movement to introduce Western European harmonization and the chorale style spread very quickly, initiating the new period of concert-like choir singing. Bortniansky, under the patronage of Catherine the Great, still remains the best example of the composer-conductors and their church choirs of the choral tradition.

By the beginning of the 20th Century there was already a great interest among Church musicians to return to the traditional roots of the canonical chant systems. Kastalsky particularly stands out among them. While choral compositions and choir singing remain popular to this day, among serious students of Church music more and more is sacred singing looked upon as a discipline of liturgical theology rather than simply as a musical art.

This is particularly so in America, as we accept the responsibility for an Orthodox inculturation of a new land, a new language and a new people. As we attempt to find our own style in response to new needs and situations (especially those of the small missions), above all we seek to be anchored to the great Tradition.

This great Tradition, however, insists neither on a rigid formalism nor a return to a hypothetically more primitive practice. There is room in Orthodox culture for both choir singing and congregational participation, for ancient chants and familiar harmonized works, as well as perhaps for new adaptations based on the timbre of the English language, developed from the local materials of our own particular time and place. All of this is possible so long as none of it contradicts our ecclesial identity as the Orthodox Church."

I also found a video showing some 4th-century-style Orthodox worship in Africa. The video features the koboro drum and the sistrum. Here is another video with some chanting/singing like you might hear in an Orthodox church. The Russians have some beautiful chanting as well, listen here. It is heavenly. It should also be noted that Orthodox incorporates all the senses, not just hearing and seeing, but also smell, touch, and taste.

I know I quoted a lot of sources, but I felt they said things better than I could. I know the original question was related to hymnody, but I wanted to make it clear that hymnody is merely a part of the Divine Liturgy in Eastern Orthodox understanding - it is not a separate discipline but an integrated part of the whole. It is difficult to fully answer this question without first discussing all aspects of Worship in the Orthodox faith (of which Divine Liturgy is only one). The hymns tie into the greater Liturgy and worship experience, which are informed by Holy Tradition and centered on Christ, Who is our Song.