Why Do My Dish Rags and Dish Towels Feel Oily?

I wash my dish rags and dish towels separately from all other laundry. I wash them in hot water and use detergent and bleach. When they come out of the dryer they feel oily . Why is this happening? They did not have any kind of oil on them when they went in the washer. My towels come out fine and all my other laundry does too. My washer is new and has never had anything oily washed in it.
I dont use liquid softener. I use dryer sheets.