I do not know if such initiative exists in the US : the "Week of prayer for Chrsitian unity" (Semaine de prière pour l'unité des chrétiens). This week, many gathering will be organized (they have already began) in France, Belgium etc between different Christian churches to pray together, to dialogue etc. This year theme is "Where two are gathered in my name, I am with them" (I apologize because I do not know the Bible translation in English). In France, the world orthodox churches will participate to these gatherings : MP (I emphasize this for the one who wants us to believe MP is not ecumenist) , Constantinople, the Romanians etc...
I noticed that on January 19th, (civil calendar which is the feast of Saint Mark of Ephesus for the new calendarists), there has been a common ecumenic meeting in a Catholic church...
Does this exist in your respective countries?