I used to listen to podcasts exclusively on my old Nano iPod, but since I acquired an iPhone (rationalized by an emergency overseas trip last spring) my trusty little iPod has been niched down to running soundtracks only.
One advantage of an iPhone for listening to podcasts is that if you're standing in one place, i.e. kneading dough, you can use the iPhone's speakers and shed the earbuds--my old iPod couldn't do that. An iPad or a laptop could be used the same way, but they take up a lot more counter space and it's harder to find a spot for them that is safe from flour clouds or water spill. I also liked that I only had to remember one device to have podcasts available to me at all times.
The only headache with using my iPhone for all my podcast listening was playlist management. I would sync my hand-picked podcast playlists (which iTunes lumps with "Music" playlists) from my laptop to the iPhone's orange Music app, or I would download podcasts from within the iPhone using the purple iTunes app (I still get confused between "Music" and "iTunes," which is why I've used chromatic identifiers here), but both methods were tedious, and gummed up storage space.
I thought the iPhone would achieve its podcast listening potential when Apple released their Apple Podcasts app in June of 2012, but then I used it: terrible search, draconically truncated titling, and non-intuitive subscription management. Half the time I can't find things I know I've downloaded. I have trouble finding things I want to delete. Most frustrating of all, I can't make playlists of podcast episodes according to my handpicked categories--i.e. "Favorite Short Stories," "Craft Lectures," "Perpetuities," or "Vermont Road Trip," something I had managed fairly simply by using iTunes on my laptop and then syncing the playlists to the iPod (N.B.: iTunes will store your playlist as "Music" whether it contains podcasts or songs, or a combo).
Now, thanks to a TidBITS article titled "Five Alternatives to Apple's Podcast App" that my favorite clipper forwarded to me, I have hope for far less frustrating iPhone podcast listening in the future. TidBITS contributor Josh Centers, who seems to share my dissatisfaction with the native Apple Podcast app, and who has a has a far more techy understanding of data management than I do, has gone to the trouble of testing five third-party podcast management apps for the iPhone. After reading his article, I'm planning to try out Shifty Jelly's Pocket Casts app (Android version also available), stat.