Are they configured to sync to your iPod Classic via the Podcasts tab? How are trying to sync them to your device? I assume no error messages during the sync operation?
To check your podcast sync configuration, plug the iPod in and select it from under the Devices section in the left hand pane of iTunes. Then locate the Podcasts tab to the right in the main window and select it. Here you can configure how and what is synced to your device for podcasts.