Naturalisation records: These may contain the date of birth, place of birth, occupation and place of residence of the immigrant as well as the name of the ship on which they arrived. They are unlikely to give a precise place of origin in Ireland. Many have been digitised by FamilySearch.org, ancestry.com and FindMyPast.com. A state-by-state, county-by-county guide to which records are online is at www.germanroots.com. Cemetery and burial records: There are two kinds of potentially valuable records: gravestone inscriptions and sextons' records. These vary enormously in usefulness but may sometimes specify the exact place of origin. Good guides are at www.deathindexes.com and www.cyndislist.com.
Immigration records and passenger lists: These are now mostly in the National Archives in Washington. The Customs Passenger Lists, dating from 1820, give only the country of origin. The Immigration Passenger Lists, from 1883, include details of the last place of residence. See here for more detail. A good online guide is at www.genealogybranches.com. Ancestry.com has the largest collection. A good one-stop search site, which covers Ancestry, is www.stevemorse.org.
Military records: Depending on place or branch of service, these may specify the place, or at least the county, of origin. First World War draft cards asked for date and location of birth and are online at www.ancestry.com. A good guide to online US military records is www.militaryindexes.com.
Church records: Marriages of recently arrived immigrants may include details of the Irish place of origin. Most Catholic records are still in the parishes. NEHGS has an excellent collection of New England Catholic records at americanancestors.org. The records of other denominations may be held locally or deposited with a variety of institutions, including public libraries, universities and diocesan archives. Good guides are at www.genealogybranches.com and www.cyndislist.com.
Biographies in county histories: Many counties in the United States have printed county histories, which can often contain biographical information about families who have lived there. The catalogue of the Library of Congress, catalog.loc.gov, can be good, as can the LDS Family History Library, familysearch.org/catalog/search.
Vital records: Death and marriage records in particular may be of value, since they generally supply parents' names. Registration began quite late in some states. See familysearch.org/wiki for details.