I have certain folders that I've restricted down to certain roles in File manager yet when a user goes to the "Link Gallery" and selects "File ( A File On Your Site )", they can see the names of all the files in the restricted folders.  Ideally, none of the restricted folders would even be displayed.  Am I doing something wrong?

Yes, I did.  I tried a couple ways with no luck:  First, I only granted permission to the role I desired to see the files.  I left everything else with an empty box.  When that didn't work, I put a red X in "All Users" but left permissions for the desired role.

I think so. I went to File Manager (as host), specified the permissions and clicked "Update". I went back to file manager later to verify that the permissions had saved. They had.

In a separate browser, I logged in as a user who should not be able to browse or open the folder. The folder appeared in the list of folders in Link Gallery. When I selected the folder, I could browse the files in the folder.

I'm running DNN 5.6.1 (238) hosted by GoDaddy.