Option 1 (use Resources module as a report)
You can create a Relations Summary field on the Resource> User object which displays the aggregated timesheet hours submitted in the week and can be seen in the Resource module. This is what we have done at another customer. It's pretty convenient.
Option 2 (weekly mail per user)
Create the Relations Summary field as mentioned in Option1 and created a Scheduled Workflow Rule to run on Friday PM per user with the aggregate hours submitted that week
You get 1 mail per user who submitted time that week
Option 3 (spam time!)
I made a workflow rule which I tested (and it worked) on the Timesheet entity with the evaluation criteria:
$State = 'Submitted' && IsMemberOf('RSS',$ReportedBy)
The downer is that you get an email for each individual time entry, which could be more hassle than it's worth (you'll get 5 mails per user and won't be able to keep tally)
Option 4 (one mail to rule them all)
In our 5.4 release coming in May you'll be able to send a report of users who submitted time.
So you could create a resource level report with the aggregate timesheets field described in option 1 and email it on a weekly basis.