In order to begin your search for Michael Mooney's family in Ireland or for anyone in Ireland or researching here to make a connection, you would need to know specifically not only what county he was born in but also the townland/civil parish within that county.
There are generally no surviving census records for the 1800s for Ireland. Those from 1821 through 1851 were largely destroyed during the Irish Civil War in 1922 when the Four Courts Building in Dublin, where they were housed, was bombarded. The 1861 and 1871 census records were destroyed by the British Government (there was no Irish Government at that time) shortly after they were completed for confidentiality reasons. And those from 1881 and 1891 were destroyed by the British Government for use as pulp during WWI.
Civil registration didn't start in Ireland until 1864 (Protestant marriages in 1845) so the source for your family's information would be church registers for the specific location, i.e. townland/civil parish/county where they resided. A description of these civil divisions is at http://www.ancestry.com/library/view/news/articles/2435.asp
A list of the names of all the civil divisions is at http://www.seanruad.com
and select any county to see the names. You would need to know also the emigrants parents names as well as their religion. LDS, the Mormons, have copies of some of the church registers and they would be available through a LDS Family History Centre.
So US records would be your source to find the specific location in Ireland for your ancestors as well as parents' names. Those records would include:
For the Emigrant: civil and religious marriage and death records, naturalisation records (esp. the Petition for Naturalisation/Declaration of Intent which provided more information than the final papers), obituaries, cemetery information, tombstones, and wills.
For the Emigrant's children: civil and religious records including birth, baptism, marriage, death, as well as the cemetery information, and tombstones.
In religious records you may also want to pay special note to the sponsors who were often relatives or neighbours from the home area in Ireland.