Certainly I am not the Patriarch of Constantinople and the courtesy of visits between the Roman Catholics and him are not something His Holiness Patriarch Bartholomew invented. Certainly Rome and Constantinople have made thousands of visits with each other before and after the schism.
So really I will attend a non-Orthodox service only under a few circumstances. If someone gets married I would gather it would be crass to skip their ceremony because it is a non-Orthodox service. If someone passes away it would be crass to skip their ceremony because it is non-Orthodox. You can attend those services and not pray physically, audibly or whatever with them. Like when Protestants say "let us pray" and everyone bows their head, well, you don't have to. Something a priest did/does at funerals I served that irked me to no end was having everyone say the Our Father together.
I would be chanting the service and he'd cut me off and say "let us pray together as one family in Christ and pray from our hearts the Our Father." He would do this knowing 80% of the people in the funeral home weren't Orthodox. He said he did it to make everyone feel included. Even then I wouldn't recite the Our Father. Not being cheeky, but saying the Our Father in that situation is wrong.
Don't we pray in the litanies "....for the union of all?" You have to stand your ground and stand up for what you believe.
While in Catholic school we were taught all religions and all denominations are various roads that lead to the same destination; God. I hear this all the time "oh it doesn't matter what church you go to everyone worships Christ in their own way." Um.... no. Study the Trinity. Study church history.
God has revealed Himself to us and the church has acknowledged how the trinity is/was revealed to us.
Other people may say they worship Christ or God but when you break it down their definition of God isn't the way it truly has been revealed to us. Guess what, you may use the words Christ, Holy Trinity. God the Father but when you mess with the theology and the revelation that has been given to us and translated and taught by the church then you in fact are not praying to the same God.
It is better to show people the Truth and the Way through love and charity and gentle teaching. Is it better for your and their salvation to pray with them and support their wrong teachings or is it better to stand firm and not pray with them and not support those wrong teachings? Praying with them is stating that you accept their false teachings.
Also the notion that all religions are good and there is one God just many paths to Him.. if you believe that as a Christian then you are denying that Christ came into the world to save sinners, that He loosed the bonds of the captives, opened the gates of Paradise and through Him Salvation rests. If you believe that there is one God, and all religions lead to that God then you are mediating Christ to a prophet.
If you are Orthodox then you believe you follow the correct teachings of the church and the revelations that God has given us. You have nothing to gain or learn by going and listening to a non-Orthodox service. If you are Orthodox then you should believe that you have the Gospel to spread as it was revealed by Christ.. you should be clamoring to have the non-Orthodox attend OUR services. We should believe that we have the greatest gift to share with the world; Christ.
If the non-Orthodox prayer groups lead us to the same place or what have you then why did hundreds of millions of people die for Orthodox Christianity? It is something to not be compromised or devalued. 2010 years means something, we exist because of God and we live for Him.