I always copy and paste the full URL of the media file ...
That is good and the attachments should also be visible in the new blog.
But there is maybe a problem with the Import plugin. I have created also a new blog to reproduce the situation. I used the Export and Import tools of WP (Import plugin version 0.5) to copy the content from an existing blog to the new one. The result is that the wp_postmeta entries of podPress (with the meta_keys _podPressMedia and _podPressPostSpecific) have no values after the import procedure. The .xml file does contain the values but somehow they get lost during the import.
I guess that is what has happened also in your case.
To correct that export and import the wp_postmeta table or maybe only the podPress entries via your database interface. (In my case it is phpMyAdmin and it worked. The posts in my new blog contain now the podPress attachments.)
Please, look into the wp_postmeta table of the new blog and search for the podpress entries. Do they exist and are their values empty?
Have you used the WordPress Importer plugin? Which version? v0.5?