Yes, they accept personal checks, postal and money orders. Make checks payable to "The Registrar-General".
The address is:
General Register Office
8-11 Lombard Street East
The fee for birth certificate is IR£5.50
Include (as many as known):
Surname, First Name(s) in full,, Date of Birth, Place of Birth, Father's Name, Father's Occupation, Mother's First Name and Maiden Surname and whether they were legally adopted.
It usually takes 6-8 weeks or more for a reply.
I have request forms for birth, marriage & death records. If interested send me a physical address or a FAX number.