It's incredible to me that this question was asked over three months ago, as of this moment, SEVEN other people have indicated that they also have a problem with this, and yet Apple can't bother to provide a solution for this issue.
I'm completely aggravated that after transferring ten podcasts from my PC to my iPhone 5S, only two show up in the podcasts app! I've double checked to make certain that the "Media Kind" is set to podcast (putting aside the fact that the setting was obviously correct because the podcasts appear in the "Podcasts" section of iTunes on the PC).
The fact that there are comments about this going back to the fall of 2013, and yet Apple has yet to address the problem, is unacceptable.
For what it's worth, though, here is my workaround. It's certainly not ideal, and it should be completely unnecessary to do these things (how much have each of us paid Apple for a functional product?), but it does allow me to listen to podcasts on my phone. I open the "Podcasts" tab on my computer, and right click (this is on a PC--is a Mac different?) on the particular podcast that I would like to sync to my phone. Click on the "Options" tab and go to the "Media Kind" pull-down menu. Change the media kind to "song." This will move the podcast to the "Music" tab. Then, after syncing, the podcast will appear in the Music app (as a song). Again, this is not a great solution, and I fear that it may create an organizational nightmare if and when Apple ever decides to provide a fix for this issue. But, it's better than not being able to listen to the podcasts at all.