This is without doubt the most common problem that I address in my working life and can occur more frequently in dogs that come from rescue centres and puppies being left alone for the first time.
This type of behaviour can produce various behavioural responses such as chewing, prolonged barking, hyperactivity and so on. Destructiveness can be intense and the worse case I have seen was with a young Boarder Collie called Tickka. She had such a wonderful disposition, but when left alone her destructiveness knew no bounds.
With the owners determination however, and a little guidance from me, she eventually became a settled and confident dog, but sadly there is not always a good outcome.
For puppies I always recommend a kennel or crate, which may sound like imprisonment but is actually a safe area or den, for your dog to spend quiet time, away from the noise of everyday life. Older dogs will also benefit from these and by placing bedding, and toys within this space, it will encourage them to return voluntarily to rest. A radio tuned to a talk station will add to his familiarity, and by leaving him for short periods, gradually increasing over time, will not be such a shock to his system.
As with all behavioural problems, the restructuring of the pack is essential and while it is tempting to assure Rover that you are going now but will be back very soon, you should in fact ignore him and just leave the room. Any prolonged goodbyes will only serve to heighten his anxiety, causing further panic when the door is closed. A good long walk prior to your leaving with also be beneficial, and an item of clothing with your scent can be a comforting reminder of his pack.
Please remember that with a little patience, and by following these simple instructions, you should be well on your way to having a well balanced dog.