The amount of times I have seen people stumped by iTunes is unreal, once you take a few minutes to figure out how it does things then its so simple
Where have you downloaded the podcasts from? did you do it directly from the iTunes Store? If not, that might be your problem.
Here's what I just did
1. Fire up iTunes
2. Go to the Podcast section of the Store
3. Search for Ricky Gervais, and Subscribe to his Podcast.
4. Let the episode download
5. Once done, plug in the iPhone, let it Sync. Once finished, unplug and go into iPod on the phone - the Gervais podcast will be sitting there for you in Podcasts
6. If it hasn't Synced, check the Sync Podcasts is ticked in the iPhone options
Unlike a lot of other audio players/phones which use drag & drop/disk mode to transfer files, iPods & iPhones don't work like that. You have to use iTunes to build up and manage your library, and when you connect your device it makes a copy of all or part of your Library Contents
Its like the classic case of people wiping the iTunes Library on their computer, then plugging in their iPod and wondering why the music on the iPod was deleted as well.....