What to use instead of Half breed?

In the show The New Normal, the kid character Shania was suspended from school because she sang the song "Half breed" by Cher as a school project.  I did not even know that the word half breed is considered offensive.   So my question now is what word to describe a person of having two racial heritage?  Mix race?  Bi racial?  In the Philippines we call anybody of mix race as Meztizo.  There are other words that come to mind.  Mullato for instance is one for half white half black people.  In Hawaii they call them Happa for half asian half caucasian.  In Vietnam they call the half Vietnamese half white as Bui Doi.  We can come up with any politically correct word to describe a person of two races but as long as it is used in a derogatory way, that word can become offensive.  So I say Boo Hoo to those people who are so sensitive about little things like the word Half Breed.  I am sure that word started out as something innocent.  Any new word they come up with will eventually become offensive if ignorant people start using them in an offensive way.  Just remember "negro" that became "colored" that became "black" that is now "African American".