Just like in the West, chant began as monophonic — just one voice, or melody. Possibly (and I say “possibly” because although this is the explanation I have always seen and it seems reasonable, I really haven’t seen anything supporting it) anyway, possibly because no instruments other than the human voice were allowed in the East, Byzantine chant developed the ison, which began as a continuous, sustained note, the base note of the tone, or mode, of the chant, ostensibly so the chanter would have a reference pitch and not drift out of tune. At some point, the original ison morphed into moving ison, where the ison or pedal tone “harmonized” with the chant melody. At this point, Eastern chant, which had been monophonic, developed into the most primitive form of homophony (a melody accompanied by chords).
Moving ison did not replace traditional ison. You still hear both.
Saints Cyril and Methodius and their missionaries took Byzantine chant to Slavic Eastern Europe, where it was nativized. Znammeny chant is held up as the original Russian liturgical music. Znammeny is traditional two-voice, chant with ison, though moving ison is more frequently heard than traditional ison.
Before we go on, all Eastern liturgical music is, like Gregorian chant, built around as system of eight tones (the octoechos), roughly corresponding to the eight modes of Gregorian chant. Eastern liturgical music has many different systems of those eight tones, however.
So Westerners out of luck? Stuck with the same old same old? Consider 'Il Canto Ambrosiano':
What we do know for certain is that a sizeable number of the oldest extant Ambrosian chants exhibit characteristics which were later expunged from or modified by the Gregorian repertory. These include: the most extensive melismatic (i.e. originally improvised ornamental passages) in all of Western chant; chant forms which were radically shortened and codified by the Franconians; the most extensive Alleluia formulae; chant formulae which predate the Franconian octoechos (8-mode) system and defy classification; ranges which far exceed the normal Gregorian chant range; melodic patterns which are much more diatonic rather than pentatonic and which can easily be sung to an ison or pedal tone; the distinctively Italianate preference for the iambic (long-short) rhythmic organization.