This is just a preliminary study presented at a recent conference, so we can't give it too much weight, but the finding is consistent with what we know about working memory, and it is of some usefulness.
The study tested the ability of young-adult native English speakers to store spoken words in short-term memory. The English words were spoken either with a standard American accent or with a pronounced but still intelligible Korean accent. Every now and then, the listeners (all unfamiliar with a Korean accent) would be asked to recall the last three words they had heard.
While there was no difference for the last and second-last words, the third word back was remembered significantly better when it was spoken in the familiar accent (80% vs 70%).
The finding suggests that the effort listeners needed to put into understanding the foreign accent used up some of their working memory, reducing their ability to hold onto the information.
The finding is consistent with previous research showing that people with hearing difficulties or who are listening in difficult circumstances (such as over a bad phone line or in a loud room) are poorer at remembering and processing the spoken information compared to individuals who are hearing more clearly.
On a practical level, this finding suggests that, if you're receiving important information (for example, medical information) from someone speaking with an unfamiliar accent, you should make special efforts to remember and process the information. For example, by asking them to speak more slowly, by taking notes and asking for clarification, etc. Those providing such information should take on board the idea that if their listeners are likely to be unfamiliar with their accent, they need to take greater care to speak slowly and clearly, with appropriate levels of repetition and elaboration. Gestures are also helpful for reducing the load on working memory.
Van Engen, K. et al. 2015. Downstream effects of accented speech on memory. Presentation 1aSC4 at the 169th meeting of the Acoustical Society of America, held May 18-22, 2015 in Pittsburgh, Pennsylvania.