Something I have done for a few departments is send them a Repeating Delivery report on Friday afternoons with their assignments for the week and the project's job number to help out.
It is an assignment report (that way you get both Issues and Tasks if you'd like and can group by Assigned To and not worry about missing anything because they are not the Task Owner).
The report not only reminds them to do their time sheet, but also reminds them of what they were supposed to be doing this week. (Now, I include not only things due this week, but also include things that are overdue in case a task from last week wasn't updated by the PM and the person did some work on it this week. But that is just me).
Let me know if that helps.