The problem in comparing the Byzantine and Roman Liturgy of the Hours is that the “Breviary” in the Roman tradition has evolved into something more akin to a private prayerbook. This is understandable since mendicant clergy needed to be able to pray the hours while they were traveling. In the Byzantine tradition, however, the Hours have always maintained the element of being the public worship of the entire Church, not just the prayer of the clergy and religious. For this reason we’ve never really developed an handy breviary that can go with us wherever we go. Instead, in order to celebrate the Hours in the Byzantine tradition, you need a small library of books:
etc., etc., etc.
Each of those, with the exception of the Horologion, is a multi-volume set unto itself.
So the short answer is, yes, the Byzantines celebrate the Liturgy of the Hours. But Nno, we do not celebrate them privately.