Today, June 26, 2015, the United States Supreme Court announced its ruling that every state must issue licenses for the marriage of same-sex couples, and every state must recognize the rights inherent in marriage of all persons who have been legally married in any other jurisdiction in the United States.
The ruling of the US Supreme Court does not change church law. Only General Conference establishes the doctrine, liturgy and discipline of The United Methodist Church.
Under the current Book of Discipline (2012) United Methodist clergy are not permitted to perform marriages or blessings for unions of same-sex couples. General Conference may decide to change that prohibition at some future point, or it may not. Our Book of Discipline also empowers bishops to be the chief interpreters of the provisions of the Book of Discipline within their Episcopal areas, subject to review by the Judicial Council if duly requested.
This means our bishops are authorized to determine in what ways UM Clergy may participate in ceremonies celebrating the marriage of same-sex couples short of functioning as the presiders of those ceremonies.
Several bishops have already indicated how they are authorizing their clergy to proceed. The lists are widely variable. Some are more and some are less specific. If you are considering how you may be called upon to participate in such ceremonies, be sure to know and follow the directions provided by your bishop.
Here is a list of links to these statements as I am currently aware of them.
I invite you to provide links to other such statements in the comments below.