| It all depends on the degree of cousinship. In the United States, marrying your first cousin is commonly seen as kind of gross, but you might not even know if someone was a distant cousin. My dad's parents were fifth cousins and fourth cousins once removed through different ancestors (this was in a rural area of North Carolina where until the 1950s, pretty much everyone was descended from the same settlers from the 1750s) and most likely no one had ever bothered to keep track of their ancestry to realize it. I think it's kind of neat that I'm my own seventh cousin.
However, actual scientific research shows that there is very little genetic risk in marrying and having children with even a first cousin, even though the fear of "babies with nine heads" (as expressed in Brighton Beach Memoirs) is usually the rationale given for forbidding these unions. The April 2002 issue of the Journal of Genetic Counseling says that children of first-cousin marriages have only a 1.7% to 2.8% higher risk of birth defects than do children of unrelated marriages. Most of the defects/diseases that do have an increased risk are already tested for in the majority of pregnancies. Nonetheless, first cousins cannot legally marry in 30 U.S. states.
Sources:
Yahoo.com, "Few Risks for Offspring of First Cousins: Panel", 4 April 2002, http://story.news.yahoo.com/news?tmpl=story&cid=594&594&e=11&u=/nm/20020404/hl_nm/genes_children_1
Willing, Richard. "Research downplays risk of cousin marriages." USA Today, 4 April 2002, http://www.usatoday.com/usatonline/20020404/3997173s.htm | 
