My 3 year old little girl loves to be around me. While it is wonderful to be loved so much, sometimes it can be a challenge when there are things that I need to accomplish by myself. If we are going anywhere, mom has to be the one who puts her in the car seat. Mom is the one who walks her down the stairs. If I go to get the mail, or take the garbage out, she wants to go with me. If I have to leave at night, I try to wait until after she is in bed. My husband says that once she realizes I am gone, she cries and cries. 
The only exception to this attachment is if she is playing with her friend. She has a friend who she often plays with, and could care less if I was around. It was perplexing to us that she would cry and throw a fit when I would leave her with my husband.
In looking for solutions, we were given some advice which has greatly helped. If you are in a similar situation, hopefully it will help you as well!
Special activities for Dad and Child around the house: We started small by him taking her outside to ride her bicycle in our driveway. In the beginning, I would look out from the window so she knew I was nearby.
Activities within short distances to home: After she became more comfortable with the bicycle riding, he took her for a walk around our block. He pointed out the different flowers and plants, which she loved.
Trips to places the child enjoys: After about a month of doing the previous 2 activities, they took a trip to McDonald's to get a happy meal. The trip lasted about 30 minutes, and she didn't cry for me once.
Start taking quick trips, leaving the child at home: I then started running 10-15 minute errands by myself. While I was gone they would read stories or he'd give her piggy-back rides.
All these little thing are building a stronger relationship between my daughter and husband. I am happy to say that it is getting easier to be separated from her. My husband enjoys the special times he has with her, she is getting closer to her dad and I am enjoying a little more freedom. I have a girl's night out coming up, and I am hopeful I will be able to leave before bedtime!