The Root of the Behavior
Chances are if they spend every moment by your side, they simply have nothing better to do. Boredom affects dogs the same way it affects humans, and idle hands are the devil's work. Try and increase the level of exercise they receive and make sure you have some toys and games around the house they enjoy. The increased exercise is important to the health of your dog and will impact the energy they have to follow you around constantly. Make sure these toys and games are things they can do without you. If you play fetch with them every time they cling to you, they will cling to you every time they want to play fetch. The same can be said for treats and bones. If you give them a treat to distract them and buy yourself some freedom you are inadvertently telling your dog to harass you every time they want a treat. It is all a vicious cycle.
Separation anxiety can be a driving force in their clinginess and there is no easy answer to this one. This can be caused by past trauma or neglect and can take years of work and comfort to overcome. Certain breeds and high strung canines will be much more susceptible to these issues. Dogs that develop disabilities later in life will often become incredibly clingy. If they lose their hearing or sight, they will become reliant on you for assistance constantly and become fearful when you are away. They will often stay very close if they can. Another sign of separation anxiety is destructive behaviors when you are away from the home. If you leave and they tear up your coach or chew up things they would not normally with you around, this can be a big indicator. Dealing with separation anxiety can be quite tricky.
Encouraging the Behavior
Creating a separate area for the dog to retreat to is essential for mental health. This can be easier then you'd think and often dog owners just use their kennel. Supplying their favorite blanket and a few toys they really enjoy can do a great deal to increase your success. Try and keep this area as far away from the highly trafficked areas of the home as possible. A puppy can have attachment issues due to something called imprinting. Though imprinting may seem like an alien term you are familiar with it in practice. When a baby duck follows a mother duck and always stay close, that bond is called imprinting. It happens in canines as well and usually dissipates with time. This is often a learned behavior. Remember not to reinforce this behavior by petting them or giving them treats when they are following you around. These best practices around your house are, over time, what will change the behavior. Often dogs end up following their human companion around when they eat simply because they are messy eaters and spill food on the floor. Essentially it can be hard to tell why your dog is following you around but once you can identify the cause of the behavior you can take the proper steps to quickly correct it.