It depends on what the job is. There are plenty of perfectly legal job descriptions that include being able to communicate fluently in English. If someone is trying to get a job that involves a lot of oral communication and a need to be understood, then yes, a thick accent could be a problem.
Honestly, if you're hard to understand you are hard to train. It's just a fact. Work on closing the communication gap. What words or letters together you have trouble with find someone to help change that.
BTW, for all of you who say that would be discrimination, I want you to know it is not discrimination if the requirements of the job include communicating with the public. Not all discrimination is illegal, if you want to know more, here is a website that explains it pretty well: http://www.is4profit.com/business-advice/employment/discrimination/illegal-discrimination.html Having said that, I love your user name, greeneyesbluesoul. Your thick accent will keep you from being able to hold some kinds of jobs, but not all of them. There are plenty of jobs that don't require you to interact with the public, and your accent shouldn't have any effect on those jobs. Don't lose heart, after being here for a while, your accent will lessen enough that you will easily find any kind of work you are qualified for without any impact from your accent. I wish you luck! P.S. Listen to lost of radio and tv to help you lessen your accent.