I doubt if you'll have a problem with your degree. If we take Saudi Arabia (SA) as an example, most websites state there is always a demand for English teachers.

You clearly have your reasons for wanting to teach in this area, probably related to the salary on offer and the fact that you may be able to save a lot of money.

I don't know how aware you are of what it is like to be a teacher there. My information is second-hand (based on what I have been told by an American I worked with in China who had worked in SA) but is confirmed in the document in the link below. You need to do the due diligence thing until you are convinced that this is the place for you.

I'm also wondering which countries in the ME you will consider - given the instability in the region as a whole? Have you spoken to or had any information from your Embassy, or their equivalent of the British Foreign Office, in your own country?

http://photos.state.gov/libraries/saudi-arabia/768382/yoderja/guide.pdf