When you enable the SSL feature and use it on a page, you will immediately get the message in IE about there being secure and unsecure items. 

"This page contains both secure and nonsecure items. Do you want to display the nonsecure items?"

I know this is because there are references to files (images, etc.) that are not using the "https" protocol.  However, what is the best solution to this in DNN?  There has to be a safe way to get this done.  I would also prefer to not have to install another module.

You'r right on with that.  I don't know about all of the Core Modules, but I was able to track this down to my inclusion of the Google Analytics script.  I had not updated it to the new one.  The new one determines if the secure version is needed or not on the fly and includes the appropriate one accordingly.  Here is a link to the Google Help page on that topic.

http://www.google.com/support/googleanalytics/bin/static.py?page=troubleshooter.cs&problem=gatsc&selected=a1h2_a1h2t2_&ctx=gatsc_a1h2_a1h2t2__55483&aw_referral=