Constipation after switching to 2% milk?

My son gets constipated ( he is 2 1/2) and the doctor said it is probably the milk ( he is on 2%). The doctor told me to limit the amount he drinks. He has 3 cups a day now. We went through him being constipated and tried a lot of different things. The doctor recommended his main drink being water throughout the day(instead of juice or milk) That is the only thing that has worked. Some days he gets more milk or more juice and sure enough he gets constipated. So I would say try water. It was the only dietary change that worked for my little boy.