1) Delete the podcasts application from your iPhone and be sure that it will not be loaded from iTunes again. (press on podcasts app until there is a minus sign for delete visible).
2) Shut down music, if it is open (double table the home button and then scroll to music and press the minus sign).
3) Power off the iPhone (hold down the top button to until a message that says slide to power off).
4) After the power off, hold down top button until apple logo appears and iphone starts.
After the above steps, podcasts were back in music and could be used with music playlists (as before).
I got these instructions for other BBS, tried them and they worked for me.
I notice that Apple has moved Podcast out of the iPod app in iOS 6. Can you still add podcast in playlists? I like to order my podcast in different order based on what I find interesting and let them play in a playlist that I create on the fly (w/out using iTunes). Also I can mix different podcast together (sports, tech, news, etc.) in one playlist and change the order or add new items as they download.