For just about every role communication skills is a standing criterion. Although content is important you can also be assessed on how well you interact with colleagues as well as potential customers.
A strong accent does not automatically suggest poor or unsuitable communication skills but it can present a barrier to effective communication, and therefore lead to you being screened out of certain roles. If your accent is so pronounced that comprehension is difficult you will find that with certain roles (for example: IT based support roles) you may not get a call for an interview.
Note that you are not being screened for accents but rather for communication skills, and there are things that you can do to lessen the impact of a strong accent on effective communication.
Firstly and most importantly,
Consider your audience. You must have regard for your audience – often employers will be taking notes, so heavily accented technical terms blurted out at a million words a minute are likely to grind their gears and result in an unfavourable assessment of communication skills. Candidates that show the self awareness to realise they have an accent and who make the effort to slow down and speak clearly are more likely to get to the interview stage.
Be aware of your environment. This goes for everyone really, but is more pronounced with strong accents – don’t call the employer from the train platform or supermarket. If your employer calls you while your child is having a tantrum or when your mobile coverage is patchy, offer to call them back, offer a landline number.
Ensure you are understood. Listen. Look for verbal nods, or clues that your content is being taken in. Ask whether your they can understand you clearly.
This demonstrates respect for your audience and this may well be the key. By making sure you are communicating effectively can ensure you will get through to the next stage of the interview and even possibly the job.