It depends.
The free template site will give you the free code and if the thumbnail image is in the code, then it's free. For instance, most free templates have many design features based on images which are intended to be free and used as part of your page otherwise there wouldn't be any point in using the template.
If the thumbnail is on a page like the template homepage that is describing what templates are available, or in a demo, or in examples, then you would be contravening copyright by using it.
Code downloaded to my PC will be deleted in due course.
WIN7; IE9, Firefox, Opera, Chrome and Safari for Windows; screen resolution usually 1366*768.
Also IE6 on W98 with 800*600 and IE8 on Vista 1440*900.