I found this great blog about how to whiten your white bed sheets without using bleach!
Here’s a quick recap of how:
Add about a half cup of baking soda to the drum of the washing machine and then add distilled white vinegar to the fabric softener dispenser.
Important to note: allowing vinegar and baking soda to combine in the process will render each of them no more useful than salt. Splitting them up, allows them each to work their magic without interference as the vinegar will get released later in the cycle.