Instead of the "Hail Mary", the last of the bidding prayers in my usual parish has been formulated along lines somewhat like these:
With the prayers of Mary, we place our own private intentions in silence before the Lord.The absence of the "Hail Mary" has not meant an absence of a Marian character to the prayer. I have not had any issue with this, generally feeling that the new practise is more reflective of the idea that elements of devotional life should not be inserted/intruded into the Liturgy properly so-called. Marian character is profoundly Liturgical; the "Hail Mary" itself a devotional prayer.
The report on this matter at The Catholic Herald, however, indicates a slightly different take on the part of some when it quotes from a parish newsletter:
Parish priests in Bishop Conry’s diocese have asked those who prepare the bidding prayers to exclude the Hail Mary. “However, this should not in any way undermine the importance of devotion to Mary and we strongly encourage the use of Marian prayers in the home and other non-Eucharistic liturgies,” said the newsletter of the parish of the Nativity of the Lord in Redhill, Surrey.The question here seems to be one of a separation of Marian devotion from Eucharistic devotion, and not one of the relation between Liturgy in its strict sense and devotions, the question that the Holy See seems to be raising if a later paragraph of The Catholic Herald report is correct .
Given the number of feasts of the Blessed Virgin that are celebrated in the Church's calendar with their proper Mass texts, and given that the Collection of Masses of the Blessed Virgin contains 46 formularies for Masses in honour of Our Lady, it appears profoundly non-Liturgical to suggest that Marian prayers and devotions should be reserved only for "non-Eucharistic liturgies" (small "l" which I take to refer to what I would consider "devotions" outside of the Liturgy, capital "L", properly so called).
Quite the contrary. The Liturgy of the Eucharist has a profoundly Marian character to it, even when the Mass being celebrated is not one of a Marian feast or a votive Mass of Our Lady. And Eucharistic devotions - adoration, benediction - should also reflect that character.