The lack of mortgage payments, repair bills, etc. is unusual and does warrant further investigation. Based on what you described, I would request a list of the rental properties and if they are local, I would have a branch manager or business development officer perform a site visit. If they are not local, I would ask the customer why they are banking at your location.
At the end of the day, if you feel the customer is structuring the deposits or you cannot verify the source of the cash as legitimate, file an SAR and consider the account for closure.
_________________________
"100 victories in 100 battles isnt the most skillful. Subduing the other's military w/o battle is the most skillful." Sun-Tzu