MailBoy periodically searches for new unread messages in your Mail.app Inbox and displays them incl. preview in an unobtrusive HUD on top of all other Apps - enabling you to decide what to do next.
Ignore them for now.
Deleting them or even mark them as junk mail.
If you like, MailBoy can be used as a Menu bar application with custom Global Hotkey and hidden Dock icon. You can customize its behavior to your needs and the shortcuts from Mail.app are available in MailBoy, too…