The Root of the Behavior
Separation anxiety is a common cause, and certainly the easiest to identify. If separation anxiety is the cause, then you'll typically not observe them destroying the pillows, rather come home to your pillow innards strewn about the home. Treating the behavior of destroying your pillows but not addressing the separation anxiety is possible through training or anti-chew sprays, but not recommended. This often leads destructive behaviors in other facets of their life, perhaps destroying shoes, toilet paper rolls, plants, etc. However, if you treat the root cause, the separation anxiety, then it will not likely manifest itself in other ways. Boredom causes dogs to act in a variety of ways, and not often pleasantly. Idle paws are the devils work as they say, they being myself alone. It just goes to say that they very well may be destroying things because they simply have nothing better to do. Finding things for your dog to do and toys they enjoy can go a long way correct this, and increasing the amount of exercise they receive each day can help deal with any excess, pent up energy they may have. In other cases, this may be a learned behavior. You, of course, never intentionally taught your dog to chew on pillows, but it can be much more innoxiously taught. Lasting impressions can be learned from the smallest things. If when they were a puppy, for instance, you came home and they had torn up a pillow and you thought it was adorable and therefore reacted that way when you came home, then it is likely the lesson they take away from that one instance is that chewing up the pillows is a good thing. Correcting learned behaviors can take a bit more work, as they will not understand why you are upset; they believe they just did a great thing.
Encouraging the Behavior
After you have identified the root cause of the behavior, you can correct it with just a few best practices. Separation anxiety can be tricky, especially if you have to spend a good amount of time away from the home. Having someone come by whenever you spend a lot of time out of the home can be an effective though costly cure. Distractions and a comfortable environment for them can make a world of difference. Toys and other things they enjoy in a place in the home that they can call their own will supply an additional amount of comfort for when they need a reprieve. Make sure this area is removed from the hustle and bustle and high traffic areas of your home. Many dog owners try and create this environment using a kennel, and to great success. It is a good way to turn the kennel into a place they love. Contacting a behavioral specialist or trainer could be a good approach to curing these behaviors, though some products such as anti-chew spray are available and can help to protect some of your things, as it will be very distasteful. Using these products have varying benefits, as some dogs react as intended to the taste while others will continue to destroy them regardless.
Other Solutions and Considerations
Dealing with dogs who suffer from separation anxiety can be tricky, as it is typically a result of other stresses in the home. Your attitude and anxiety can transfer over to your dog as they empathize extremely well. Try to remain calm and relaxed around your dog, as it will reduce their general anxiety. In younger dogs, this is in general to relieve pain from the teething process. This will typically pass when they have grown all their teeth. Chew toys can often provide an alternative for them to chew on, and removing the chew toy in later years will be much easier than removing all your pillows.
Providing an alternative thing to chew, whether antler or chew toy, will provide some immediate relief from destructive behavior, will still encourage chewing as a general behavior. Removing the chewing fixation will often happen with time, additional exercise, and other forms of entertainment. Reducing the anxiety in your life and household will make a large difference to the general anxiety of the dog, even when you are removed from the home.