You can use dynamic named range for the source data :
exemple for data in columns A to D :
Insert / Name
Name : myRange
Refers to :
=OFFSET(Sheet1!$A$1:$D$1,,,COUNTA(Sheet1!$A:$A))
Have a look at the exemple file :
http://www.filedropper.com/excelpivottableproblem
HTH
Daniel

I create pivot tables in a time tracking worksheet I use each month. However,
when I go back to a saved worksheet and enter new data, I can't refresh the
pivot tables. Therefore, I have to re-build the pivot tables in each
employee's worksheet at the end of each month.

What am I doing wrong? :)