How can I get my beige bath rugs cleaned?

I have put these in front loading washing machine and have never had a problem

I have also washed mine in the washer. Just don't use the hottest setting on either the washer or dryer.

Yes they can, just wash cold on gentle cycle with minimal detergent. Lay flat to dry on beach towels or old towels or blankets. I have several sets and wash mine once a week but I have a bunch of pets so my stuff gets washed far more often than normal people.

I've also washed mine in the machine.

If you wash them in the washing machine, do not use bleach, as it will deteriorate the rubber backing, and dry them on a clothes line, not the dryer.

I wash mine on the driveway, pour a bucket of water with laundry detergent over the rug, let sit an hour, rinse with garden hose. Dry by draping over the fence!