Are you using the gradebook properly? While I am not a big fan of notifications they do come in handy. Also, the Teacher of a course should know what their Students have done and not done, or do you have a collective marking system? Who is the person most responsible for a course? The contact person, the tutor, the course Manager, call him/her whatever you like. Each course has to have someone responsible for that course. That person is the one who should be on top of these things. I am assuming you have a school-like setup with a 5 to 1 ratio of courses to Teachers. If you have a business structure, that is different, you need a bit more help than the above.
Assuming you have server access, I seriously suggest you install phpMyAdmin and access the Moodle database. If you look at the mdl_assignment_submission table and perhaps run a query that looks something like
SELECT 'assignment','timecreated','userid','grade' FROM 'mdl_assignment_submissions' WHERE 'grade'= " "
Obviously I have no idea what your grading standards are, so you might want to use 'grade' = 0 or 'grade' <= " " or similar. [My SQL is as rusty as a dunny door, so don't take my word for it here, but use phpMyAdmin to guide you in creating the statement.]
You might be able to generate a report using the AdHoc Reports or Configurable Reports plugins.