The Church does not prohibit interracial marriage. Unlike marrying someone from a different religion, (See Dating and Marrying Nonmembers), it is not considered a sin to marry someone of a different race. That said, church leaders have expressed concern about any circumstances that would make a marrage more stressful. This could include couples from dramatically different cultural backgrounds (President Spencer W. Kimball, Quoted in FARMS, Review of Books on the Book of Mormon, Vol.7, Number 2, p.68). Members of the church who ignore this counsel are not penalized or ostracized.
In the end, it is up to each individual to decide whether the cultural differences inherent in interracial relationships are significant or not in the context of their relationship. The answer will vary by culture and the people involved. For example, in Bosnia, an Ethnic Albanian and a Serb who married would unlikely be accepted by either ethnic group. Furthermore, the couple and their children might even be targeted by mob violence and death. On the other hand, the exact same couple living in a more tolerant location (e.g., Sweden) might not face any repercussions at all. Likewise, a liberated American woman might have strong conflicts with a husband from Afghanistan who strongly espoused the views of his culture towards women, but might get along very well with a man from the same origin who did not strongly hold those views. Finally, a mixed race couple from one culture (e.g., a White man and a Black woman from the same small town in England) might get along better than two people of the same ethnic stock that were raised in different cultures (e.g., a woman from the American South with Dutch ancestry who married a man from the Netherlands).
Cultural backgrounds are only one of many factors to consider in choosing who to date and who to marry. They are, however, important so consider thoughtfully as culture often has more influence than many couples may suspect. Mature individuals consider the possible difficulties very seriously before dating and falling in love. When you are in love, it seems like you can overcome anything. It is important to make sure that racial and cultural differences don't become a stumbling block to a successful marriage after the initial infatuations have worn off.