Am I right in thinking that the Western concept of Advent does not really exist in Orthodoxy?
Something of the Western concept of Advent exists within the tradition of Syrian Orthodoxy.
For us, the new liturgical year starts on the First Sunday of Advent, but that Sunday is called the Sunday of the Dedication of the Church, and is usually the first Sunday of November (or the last Sunday of October, if the date is 30 or 31 October). The next Sunday is the Sunday of the Renewal of the Church.
After these two Sundays, we start a series of Sundays which are special for their Gospel readings. On these Sundays, various portions of the Gospels relating to the events before the Nativity are read. So, in order, we have the Sunday of Zechariah (on which is read the Gospel about Gabriel's visit to Zechariah to announce the conception of John in his mother's womb), the Sunday of the Annunciation (the visit of Gabriel to the Virgin Mary), the Sunday of the Visitation (Mary goes to visit Elizabeth), the Sunday of the Birth of John the Baptist, the Sunday of the Revelation of Saint Joseph (commemorating the dream Joseph has in which the angel tells him to go ahead and marry Mary), and finally, the Sunday before Christmas, on which we read the genealogy from Saint Matthew's Gospel. After the Sunday of the Annunciation, the Divine Office (Horologion) has daily propers for the weekdays "in the time of the Annunciation".
The overall effect of the fasting and the cycle of Sundays, especially if in conjunction with the daily Offices, is to really create a sense of intense expectation that lasts a good two months. Christmas becomes all the more special this way, for me. I've found that Western Advent does not even come close to this, although (and I'll say right now I have very limited experience with Eastern Orthodox practices) I am of the opinion that, with the exception of the fasting, Western Advent is probably a better thing than the EO preparation for the Nativity.