It is fantastic that you are researching your Griffith ancestors! It is also very helpful that you already know where they came from.
Before 1864 records were kept in the form of church and parish records; while some of these have regrettably been lost over the years, all the surviving records have been sorted into county-based genealogical centres
. In your case, the genealogical centre would be the Cavan Genealogy Centre
Because you know the place they were born at, you can also search the Church records
on the Irish Genealogy website.
Have you been in touch with the Cavan Genealogy Centre yet?
I hope this is helpful, please let us know how the search goes.