I was having the same problem, but it is now working OK.
My site is only visited occasionally, so, as suggested in this forum, I setup a cron job (on the https://www.setcronjob.com website) to access my homepage every 30 minutes, but this did not work either.
The helpful staff at https://www.setcronjob.com had a look at it for me and corrected the cron job to directly call the get_mail.php page on my server using the following URL (replace “MyWebsite.com” with your own):
https://MyWebsite.com/wp-content/plugins/postie/get_mail.php
Now it works great. There may be other ways to get it to work, but this does the job for me OK.