It turned out, this reader listened to podcasts directly from the web. Web browsers call an embedded player to run the file on the fly and as a consequence, they will not remember the playback position. Remember Playback Position is a default setting you will get when you subscribe to podcasts in iTunes. If however, you are playing a direct download, the files will not have this setting. You can, however, change it.
Select any file in iTunes you want to be able to listen to and continue where you left off afterwards. Right click the file and choose Get Info.The info of the file will pop up. Here you have several tabs, one them being Options. Select Options and then check Remember Playback Position. Click OK and you are done.
Once this setting is chosen in iTunes, it will also work once you have transferred the file to iPod. Other players and other podcatcher must have various ways to achieve the same effect.
More basic instructions:
Preference settings per podcast,
Subscribing to podcasts in iTunes,
Install iTunes,
downloading audio files,
Listening on-line.
