Do you have any dates or locations? They will probably help you get a better response.
FamilySearch.org have many of the Irish Birth/Marriage/Death Resgistration indexes transcribed online. Maybe you could order your grandmothers birth certificate, which should give you info about her parents.
For example, there's the birth of a Margaret Ethel Glavin listed in the Cork area in the September quarter of 1923.
Thank you very much, I will try that.
The Mary Ethel you mentioned would be too young to be my grandmother but I do know they came from County Cork.