You might also be interested in using perl's wonderful glob features. They allow you to do nice things like:

foreach $old_name (glob("*.DOC")) {
  my $new_name = &add_date($filename);
  rename($old_name,$new_name);
}
[download]

This executes the foreach loop for each value of $old_name where $old_name is a filename matching "*.DOC". Note that you need to be in the correct directory to get the matches.

&add_date would include the other chunks of code that other respondants have kindly written for you.

eg.
sub add_date { return s/\.DOC/time().".DOC"/e; }

(which I haven't tested)