In your instance's email notification options, you have a handful related to timesheets. Possible for you to temporarily turn those off in Setup while you're closing out your old ones? Just know if you do this and then turn them back on, I believe anyone who has turned those off in their own profile will have them turned on again after you re-turn them on (someone should confirm this).
BUT, if you do not need these really old timesheets at all anymore and don't need record of this historical time (you may need it), you could create a timesheet report for the timeframe you need, then bulk select them all and delete...
