I am an American teaching in New Zealand and haven't found it difficult at all. You will however have to either take a lower wage or complete a one year, post graduate teaching course in the New Zealand curriculum based on the area you would like to teach (Primary, secondary or early childhood).
Someone has already posted the Teach NZ website I believe, which is the best resource for teachers in NZ.
One of the main things to keep in mind is that not only is NZ a multi-cultural society, it is firstly a bi-cultural society, made up of Maori (the indigenous people of Aotearoa) and Pakeha (NZers of European decent). The Maori culture is an important part of what makes NZ unique and beautiful and taking time to learn about the culture will help to make your transition more enjoyable.