Unfortunately, this is a problem faced by all non-native speakers and while many excuses are made, eg pronunciation, the truth is that it's pure discimination. There are, though, many schools out there who don't disciminate and these are nearly always the best ones. So while it may be more difficult to find a job, when you do it will probably be at a good school.
Also remember the qualities you bring to the job: A knowledge of grammar which few native speakers, at least when they start an EFL career, will have, pronunciation which is probably better than native speakers (at least better than most of my native speaking friends) and a knowledge of the target culture. Most important of all you have already gone through the process of learning English and are aware of some of the difficulties your students will face (and how difficult learning a foreign language can be). I often wonder how many native speakers would be able to pass the Cambridge Proficiency Exam and I suspect supprisingly few.
I've been posting at this site on this subject for several years now so if anyone has any ideas about how to eradicate this stupid modern myth (that native speakers are best) from the teaching profession please, please let me know.
Good luck with your job hunting.
Kevin (native speaker)