It's nice to be loved, but when your new puppy loves you so much it cries every time you leave the room, it can wear thin. A puppy's whining at being left alone is a sign of separation anxiety, which occurs especially when a puppy is moved into a new home or with a new family, if its routine has significantly changed, or if it has had trauma in its young life. If the separation anxiety isn't severe, you can train your puppy to remain calm when you leave the room. Seek an animal behaviourist if your puppy doesn't improve or seems to get worse.
Acclimate your puppy to spending time without you. Close the door between you in one room and your puppy in another, or use baby gates to keep your puppy separate from you for several periods of varying lengths every day.
Ignore your puppy when it cries. Don't make eye contact and don't pick it up or pet it. It will associate crying with these rewards, and that will encourage the behaviour.
Reward your puppy when it is calm and relaxed. Take it a treat when it has waited calmly for you in a separate room.
Keep special treats on hand for when you will be gone for a long period of time, whether in another room or out of your home. Many dog toys are designed to hold snacks inside that require effort and time for your dog to get to. Leave one of these with your puppy when you leave it alone -- and only when you leave it alone -- and it will associate this pleasant reward with your absence.
Train your puppy to sleep somewhere other than your bedroom. This will help it become more independent.
No matter how anxious your puppy seems, never give it any kind of calming medication without the supervision of a veterinarian.