The tradition of prayers at specific hours of the day is one that goes back as far as the ancient Hebrews.
Seven times a day have I praised Thee for the judgments of Thy righteousness. —Ps. 118:164 LXX
There are also a few examples of this tradition in the book of Acts, among the early apostolic Christians:
Now Peter and John were going up together into the temple during the hour of prayer, the ninth hour. (3:1)
And on the morrow, as they sojourned and drew near to the city, Peter went up on the housetop to pray, about the sixth hour. (10:9)
The Didache or Teaching of the Twelve (ca. A.D. 60) instructs Christians to pray the Our Father or Lord’s Prayer three times a day.
Pliny the Younger (A.D. 63–117) wrote to the Emperor Trajan:
[T]hey met on a stated day before it was light, and addressed a form of prayer to Christ, as to a deity. —Epistulae, Book 10, Letter 97
Pliny is, incidentally, describing what sounds like a Matins or Othros (morning prayer) service of the Orthodox Church today, which is then followed by the Eucharistic liturgy itself.
On praying at particular hours, Schmemann notes:
[I]n the Epistle of Clement of Rome to the Corinthians we read: ‘We must do all things in order … at fixed times … not haphazardly and not without order, but at definite times and hours.’ Three hours of prayer are indicated in the Didache, by Tertullian, by Cyprian of Carthage, by Origen, in the Apostolic Tradition of Hippolytus. ‘We should pray in the early morning,’ writes Cyprian, ‘that by means of our morning prayer the resurrection of the Lord might be recalled; also at the setting of the sun and in the evening we should pray again …’ The tradition of hours and times of prayer can certainly be accepted as a tradition common to the whole of the early Church. —Introduction to Liturgical Theology, p. 82
These hours of prayer can be observed privately, in a monastic community (where they are most prevalent today), or even in a local parish setting. The Scriptural psalms play a major part in these prayers, as well as prayers particular to the day Christ was crucified (e.g. at the ninth hour, a prayer references the sky going dark).
Ultimately, the ancient emphasis was on praying together; often, in faithful unity, and in an orderly manner:
Take heed, then, often to come together to give thanks to God, and show forth His praise. For when you assemble frequently in the same place, the powers of Satan are destroyed, and the destruction at which he aims is prevented by the unity of your faith. —St. Ignatius of Antioch, Ephesians 13