They have always been sung, historically speaking. This is indicated by, among other things, the practice of all ancient Christian churches and all branches of Judaism including the more obscure Karaite and Beta Israel (Ethiopian) Jews, and also by the preface to the Psalms themselves, which are addressed in many cases “to the court musician.” Indeed they were accompanied by musical instruments at one time, which are specified for some of the Psalms.
Additionally most Protestant denominations historically sang the Psalms, many of them using them exclusively (in the case of early Baptists, Puritans and Calvinists) or primarily (in the case of the Church of England - Anglicans and many English speaking Orthodox use the Coverdale Psalter or the Coverdale-based Jordanville Psalter (which has the Psalms corrected according to the Septuagint) as it is superior for singing than the version in the King James Bible, which is nonetheless exquisite, particularly Psalm 23.
Now many Christian monks did speak them or recite them silently, and in Coptic Orthodox churches in addition to some Psalms being sung during the Liturgy and the Psalmody, there are also the Hours where the Psalms proper to each of the canonical hours (the 1st, 3rd, 6th, 9th, 11th, 12th and an extra prayer mainly used by clergy and monastics called “the Prayer of the Veil” are assigned to members of the congregation to read silently. Typically the 1st, 3rd and 6th hour are prayed as the first part of morning prayers before the Divine Liturgy (the main Holy Communion service with four Scripture lessons and a sermon).