After watching "My big fat Greek Wedding" and comparing Toula and Ian's cultures I started thinking about how popular and common weddings between people from different cultures have became
Now that we are living in a globalized world, communication and relationships between people from differents countries are topics that are seeing as normal and natural in society, but that doesn't mean there ain't problems that affect these type of  commitments.
One of the biggest problems is the perception of the family, some families (depending the culture) are very reserved and close when people from different groups referred. In this case, some of Toula's siblings were skeptics about her relationship with Ian, they considered that he wouldn't gave her a proper life for not being greek besides they didn't respect Toula's privacy; Ian's parents were most of all afraid of Toula's familly, they were overwhelmed and surprised by there manners.
Another problem is religion, people all over the world have different ways to celebrate their weddings, holidays, etc, acording their believes and spiritualities; although Toula and her family lived in North America (they are a subculture, Greek- American), they kept greek values an believes, that means they celebrate weddings acording their greek roots, that caused a problem for Ian beacuse he was a middle- class American, so he had to baptize as a greek.
I haven't married  yet so I can't give a reliable source, but for what i've seen on movies and friends experiences  those are the most common and difficult problems that could arise between marriages from different cultures.