WORKAROUND: In iTunes, select the podcast files and hit CMD-I (show info). Set the genre to something other than "Podcast", and in the Options tab, set the Media Type to Music. You can do this to the files that are on your iPhone. If you do this to the files on your computer, they will no longer show up as Podcasts - they'll be mixed in with music files in your library. You can then put these podcasts into playlists on the iPhone, and they'll show up on the phone once it's disconnected. You might need to fool around with this a bit - try it more than once, etc. - I had some podcasts not show up on the iPhone until after a SECOND connect-and-disconnect operation with the computer. Others here might do some experimenting and figure out a smoother set of instructions that will always work - but this is a starting point.
Note that I ALWAYS "manage music and video files manually" - I NEVER "sync" music and video. Your mileage may vary.
Apple - please listen: The handling of podcasts in iOS 7 is not a feature, it's a bug. As you can tell from the comments here, MANY people need to manage podcasts as if they were music files (and sometimes, THEY ARE!!!!). The arbitrary decision to treat podcasts differently is WRONG - an audio file is an audio file. I don't need to buy one TV to watch sports and one TV to watch news - because video is video. Get it? If you want to provide a Podcasts app, that's fine, and anyone who wants to use it can do so. But don't RUIN the previous, excellent functionality that we had before, when it's completely unnecessary and a colossal pain (not to mention insult) to your customers. And in some cases, it's a serious driving safety issue.
I'll be watching this space to see if others can refine the basic workaround that I've found.
UPDATE: It appears that changing the genre is unnecessary, only the "Media Type". It's probably best to do this only to the copies on the iPhone, so the files in your computer's library are not altered.