This is Life!: Revolutions Around the Cruciform Axis

The Indiction of the Ecclesiastical Year

Happy New Year to all my Orthodox brothers and sisters!

Troparion of the New Year Tone 2
Lord of the universe who by Thy power hast established the times and seasons,/ bless this year with Thy goodness,/ preserve our rulers and keep Thy flock in peace,/ through the prayers of the Mother of God, and save us.

Kontakion of the New Year Tone 4
Creator and Lord of the ages,/ God of all, transcendent in essence: Bless this year./ Save all who worship and cry:/ Grant us a fruitful year,/ O Compassionate Redeemer.

Fr. Michael Harper here explains the Orthodox New Year:

It can be frustrating to move suddenly from the end, back to the beginning of something. But this is what Orthodox believers do as we move from August 31st - the last day of the old year, to September 1st, the first day of the new year.

It is part of the goodness of God, that He, who has no beginning and no ending, the Eternal Trinity, should take such care to give us a year which begins and ends, and then begins all over again. In our human and finite state we need fresh starts, and this is one of them. From the peaks of Pascha, Ascension, Pentecost, and Transfiguration, we move back to beginnings, the Nativity of the Mother of God, and then in December of the Son of God Himself. We start this wonderful cycle all over again. But the Holy Spirit, as we trust Him, will renew this new year to us, and give us a whole new understanding of it. . . .

It is significant that the last great feast of the old year is that of the Dormition of Mary, the Mother of God. Her human passing was to heaven's glory. And the first great feast of the new year is her Nativity. It is not that Mary is more important that Christ, around which most of the Calendar revolves. Mary is not God. She did not exist from eternity. But she is honoured in this way because she is our supreme example. She lived a life of complete obedience to God.

Another explanation is given here.