Despite repeated appeals from most of the Communion, the US Church has continued its break with tradition by consecrating partnered homosexuals as bishops and blessing same-sex couples. Most recently, two lesbian Episcopalian priests were married by the Bishop of Massachusetts on New Year's Day.
Explaining their decision to stay away last November, the Primates said that liberal parts of the Communion had "torn the fabric of our life together" and that they could therefore "no longer maintain the illusion of normalcy".
In an interview with BBC Radio Ulster on Sunday, Anglican Communion Secretary General Canon Kenneth Kearon said absent Primates were still committed to the Anglican Communion.
"Those Primates who said they're not coming as part of an objection to the Episcopal Church and other developments have reiterated their commitment to the Communion and the Archbishop of Canterbury in their writing to me," he said.
Canon Chris Sugden, of orthodox group Anglican Mainstream, defended the decision of the Primates not to attend the meeting.
Writing in the latest edition of Evangelicals Now, he said the Primates' absence called into question the Archbishop of Canterbury's ability to fulfil his role as gatherer of the Communion.
While some Anglicans say the Primates should attend so that they can present their views, Canon Sugden said that such an argument was "flawed".
"It is of course true that all Primates, whether of conservative or liberal persuasion, should present the views of their province as a whole as well as their own," he wrote.
"But the real problem is that all the decisions made at previous meetings … have been ignored, undermined or overturned.
"There are only so many times you can appeal to people to turn up and make their voice heard. When it becomes clear that what that voice has achieved through turning up has been ignored, then it is unwisdom to expect anything different the next time.
"The Primates are waiting for decisions they have already taken part in (beginning from Lambeth 1998) to be respected and honoured."
There is no fixed agenda at the Primates meetings and there are no formal votes on resolutions. The meetings are held rather to agree on guidance and set the general direction of the Communion.
The outcomes of the Dublin meeting will be made public when it ends on Sunday.