Scandinavians tended to settle in groups or even form their own communities. This made it easier for them to preserve their culture and to get along in a new country amid strangers. To continue their traditions and religion, they would often find someone who had been trained in the Lutheran clergy to lead their religious meetings and who also was responsible for recording the vital records for them. This means you will want to search for church records, because you can expect them to be just as good as the ones found in Scandinavia. At times families also kept their own records of births, marriages, and deaths in a family Bible, so those can also be good sources for vital information.
The Scandinavians established their own newspapers in various places so they could keep up on the information that pertained to them especially. These ethnic newspapers contained much of the same types of information that can be found in American papers of that time period. They might also include other information, such as people looking for relatives, which can give clues to family relationships. The three main Scandinavian newspapers published in the United States were Decorah-Posten, Minneapolis Tidende, and Skandinaven, although there were several smaller ones that served the different communities.
Scandinavians generally took part in the American experience completely. Many were naturalized, so there are records from that process that can be obtained, and several fought for their new country. The Civil War was one in which there were whole units comprised of Scandinavians. Websites exist dedicated to these units, and most contain biographical information on the participants, which may include their place of origin in the homeland. One of these sites is http://www.15thwisconsin.net/index.html#Latest. The veteran records from these wars will also include information that will give more of the history of your ancestor and may include genealogical information.
Another place that could help in the search is town and county histories. If one exists for the place your ancestor settled, it would likely include biographies and histories of individuals and families who resided there, which may include your own.
One source that can be overlooked is old letters. It was important for Scandinavians to keep in touch with either those they left behind, or those who decided to settle in a place far from their own. Many of these letters still exist, whether in library archives, or in private hands. These letters can give clues to the everyday events or even to where they came from. Do not be surprised to find them written in Danish, Norwegian, or Swedish, for that is how they could communicate most easily. When trying to determine whether any exist, you will want to locate libraries that specialize in Scandinavian studies or history, or look to your family members, whether closely related or not. You may also want to consider any friends from the homeland which shared the emigration experience, and look to their descendants to see if any communications were exchanged between them; you may find clues as to who these friends may have been through letters that are in your own possession.
In addition to these sources, you will also want to look for old photographs, obituaries, probate records, land and property records, printed genealogies, civil vital records, passenger arrival lists, census records, and cemeteries. As you search you may want to consider the reason they immigrated and look for records generated by those organizations. As an example, there were many who came as members of the LDS church so you will want to search the records they may have. A good place to start there is the LDS Mission Index (FHL Fiche #6060482).
These sources and others can help you find your Scandinavian ancestor's origin. Their historical details can also provide the scenery for your journey along the road they pave across "the pond."