This is a sad account of the decline of Psalms in the western liturgy taken from the 1913 edition of the Catholic Encyclopedia under the entry for "Gradual". I have taken the liberty of breaking it up into paragraphs and deleting the source citations for easier reading.
Gradual, in English often called Grail, is the oldest and most important of the four chants that make up the choir's part of the Proper of the Mass. Whereas the three others (Introit, Offertory, and Communion) were introduced later, [to] fill up the time while something was being done, the Gradual (with its supplement, the Tract or Alleluia) represents the singing of psalms alternating with readings from the Bible, a custom that is as old as these readings themselves. Like them, the psalms at this place are an inheritance from the service of the Synagogue. Copied from that service, alternate readings and psalms filled up a great part of the first half of the Liturgy in every part of the Christian world from the beginning.
Originally whole psalms were sung. In the "Apostolic Constitutions" they are chanted after the lessons from the Old Testament: "The readings by the two (lectors) being finished, let another one sing the hymns of David and the people sing the last words after him." This use of whole psalms went on till the fifth century. St. Augustine says: "We have heard first the lesson from the Apostle. Then we sang a psalm. After that the lesson of the gospel showed us the ten lepers healed."
These psalms were an essential part of the Liturgy, quite as much as the lessons. "They are sung for their own sake; meanwhile the celebrants and assistants have nothing to do but to listen to them." They were sung in the form of a psalmus responsorius, that is to say, the whole text was chanted by one person — a reader appointed for this purpose. (For some time before St. Gregory I, to sing these psalms was a privilege of deacons at Rome. It was suppressed by him in 595.) The people answered each clause or verse by some acclamation. In the "Apostolic Constitutions" they repeat his last modulations.
Another way was to sing some ejaculation each time. An obvious model of this was Ps. cxxxv [Hebrew: 136] with its refrain: "quoniam in æternum misericordia eius" ["for his mercy endures for ever"]; from which we conclude that the Jews too knew the principle of the responsory psalm. . . . It appears that originally, while the number of biblical lessons was still indefinite, one psalm was sung after each.
When three lessons became the normal custom (a Prophecy, Epistle, and Gospel) they were separated by two psalms. During the fifth century the lessons at Rome were reduced to two; but the psalms still remain two, although both are now joined together between the Epistle and Gospel, as we shall see. Meanwhile, as in the case of many parts of the Liturgy, the psalms were curtailed, till only fragments of them were left. This process, applied to the first of the two, produced our Gradual; the second became the Alleluia or Tract. . . .
It is difficult to say exactly when the Gradual got its present form. We have seen that in St. Augustine's time, in Africa, a whole psalm was still sung. So also St. John Chrysostom alludes to whole psalms sung after the lessons. . . . In Rome the psalm seems not yet to have been curtailed: "Wherefore we have sung the psalm of David with united voices, not for our honour, but for the glory of Christ the Lord." Between this time and the early Middle Ages the process of curtailing brought about our present  arrangement.
One of the things the 16th-century Reformers wished to do was to restore the Psalms to worship, an effort that appears to need renewal every generation, even in churches that are heirs to the Reformation. My own website, dedicated to the Genevan Psalter, is intended to be part of this effort.