From the Catholic Encyclopedia:
Septuagesima is today inaugurated in the Roman Martyrology by the words: "Septuagesima Sunday, on which the canticle of the Lord, Alleluja, ceases to be said". On the Saturday preceding, the Roman Breviary notes that after the "Benedicamus" of Vespers two Alleluias are to be added, that thenceforth it is to be omitted till Easter, and in its place "Laus tibi Domine" is to be said at the beginning of the Office. Formerly the farewell to the Alleluia was quite solemn. In an Antiphonary of the Church of St. Cornelius at Compiègne we find two special antiphons. Spain had a short Office consisting of a hymn, chapter, antiphon, and sequence. Missals in Germany up to the fifteenth century had a beautiful sequence. In French churches they sang the hymn "Alleluia, dulce carmen" (Guéranger, IV, 14) which was well-known among the Anglo-Saxons (Rock, IV, 69). The "Te Deum" is not recited at Matins, except on feasts. The lessons of the first Nocturn are taken from Genesis, relating the fall and subsequent misery of man and thus giving a fit preparation for the Lenten season. In the Mass of Sunday and ferias the Gloria in Excelsis is entirely omitted. In all Masses a Tract is added to the Gradual.
From the website of the now-defunct Holy Trinity German parish in Boston:
As Septuagesima (Latin for "seventy") is seventy days before Easter, it typologically commemorates the seventy years of exile spent by the Jews in Babylon. As Psalm 136 attests, God's chosen people did not deem it fit to sing their joyous songs from Sion during the Babylonian exile, and neither do Catholics during theirs. The joyful "Alleluia" is thus laid to rest for seventy days until it rises again in the Easter Vigil. As mentioned elsewhere, this dismissal, or depositiio, of the Alleluia can take place formally in a special ceremony. After the Saturday office of None or at some point of the afternoon on the day before Septuagesima Sunday, the choir gathers in the church where it carries a plaque or banner bearing the word "Alleluia" through the church as it sings the touching hymn, "Alleluia, dulce carmen" (part of which is quoted elsewhere). It is then solemnly "buried" in some place in the church. In the Middle Ages this procession could become quite elaborate. Sometimes the "Alleluia" plaque would be in the shape of a coffin, while in parts of France, a straw man with the word "Alleluia" was even burned in effigy in the churchyard. A simpler ceremony based on the same principles, however, can easily be held in one's home or parish.
And finally, from Fr. Franz Xaver Weiser (as quoted on the Canberra TLM blog):
The depositio (discontinuance) of the Alleluia on the eve of Septuagesima assumed in mediæval times a solemn and emotional note of saying farewell to the beloved song. Despite the fact that Pope Alexander II [in the eleventh century] had ordered a very simple and sombre way of "deposing" the Alleluia, a variety of farewell customs prevailed in many countries up to the sixteenth century. They were inspired by the sentiment that Bishop William Duranti (1296) voiced in his commentaries on the Divine Office: "We part from the Alleluia as from a beloved friend, whom we embrace many times and kiss on the mouth, head and hand, before we leave him" [a reference to the Rationale divinorum Officiorum of William DURAND, or DURANDUS, Bishop of Mende, 1230-96].
The liturgical office on the eve of Septuagesima was performed in many churches with special solemnity, and alleluias were freely inserted in the sacred text, even to the number of twenty-eight final alleluias in the church of Auxerre in France. This custom also inspired some tender poems that were sung or recited during Vespers in honour of the sacred word. The best-known of these hymns is Allelúia, dulce carmen ("Alleluia, Song of Gladness"), composed by an unknown author of the tenth century [...]
In some French churches the custom developed in ancient times of allowing the congregation to take part in the celebration of a quasi-liturgical farewell ceremony. The clergy abstained from any role in this popular service. Choirboys officiated in their stead at what was called "Burial of the Alleluia" performed the Saturday afternoon before Septuagesima Sunday. We find a description of it in the fifteenth-century statute book of the church of Toul:
"On Saturday before Septuagesima Sunday all choir boys gather in the sacristy during the prayer of the None, to prepare for the burial of the Alleluia. After the last Benedicámus (i.e., at the end of the service) they march in procession with crosses, tapers, holy water and censers; and they carry a coffin, as in a funeral. Thus they proceed through the aisle, moaning and morning, until they reach the cloister. There they bury the coffin; they sprinkle it with holy water and incense it; whereupon they return to the sacristy by the same way."
In Paris, a straw figure bearing in golden letters the inscription "Alleluia" was carried out of the choir at the end of the service, and burned in the churchyard [...]
Thus the Alleluia is sung for the last time, and not heard again until it suddenly bursts into glory during the Mass of the Easter Vigil, when the celebrant intones this sacred word after the Epistle, repeating it three times as a jubilant herald of the Resurrection of Christ.