I think that the public praying of the Divine Office will go a long way in ecumenical relations. I've often thought about how awkward things are for Catholics when we ask non-Catholics (or fallen away Catholics) to come to Mass with us. After all, we hear that a lot, don't we? "Invite someone to Mass!" And then what happens? You get to the Eucharist and have to tell the person you just invited, "Hey, you're not allowed to come with me - just sit there for a few minutes. I'll be right back." Now, some people may not care, but my Mom is someone who feels very insulted by this. Theology doesn't work. Scripture doesn't work. Logic doesn't work. No matter what I say, she still feels like the Church is denying her Jesus and oftentimes she won't go with us to Mass because it upsets her that she cannot receive the Eucharist.
Because Catholics believe that the celebration of the Eucharist is a sign of the reality of the oneness of faith, life, and worship, members of those churches with whom we are not yet fully united are ordinarily not admitted to Holy Communion.
It can certainly be awkward. You can remove that awkwardness by encouraging ecumenical prayer services; the popes have lead by example in this regard. For instance, Pope Benedict XVI took part in Anglican and Lutheran prayer services, and their leadership took part in Catholic prayer services. We unite with one another in our love of Christ and the Scriptures, along with listening to one another in our preaching. At the end of the day, we're striving to break bread together:
What has revealed the love of God among us is that the Father has sent into the world His only-begotten Son, so that, being made man, He might by His redemption give new life to the entire human race and unify it. Before offering Himself up as a spotless victim upon the altar, Christ prayed to His Father for all who believe in Him: "that they all may be one; even as thou, Father, art in me, and I in thee, that they also may be one in us, so that the world may believe that thou has sent me". In His Church He instituted the wonderful sacrament of the Eucharist by which the unity of His Church is both signified and made a reality (UR, 2).
Until that day arrives, we can only benefit from praying with and for one another. Sadly, at least in the dioceses of Metuchen and Trenton, you cannot find much public praying of the Liturgy of the Hours. While the Second Vatican Council encouraged every parish to offer Morning and Evening Prayers in public, encouraging the public to take part, it hasn't really happened. I think this is a missed opportunity: 1) the laity continue to be left in the dark about the daily office, and 2) we still don't have that public prayer time where we can invite someone who may not have been to church (for a while or at all). I think this is something we must fix immediately.