Here are some instructions to downgrade your iOS back to iOS 6, which is the only way I've found to solve this issue:
0. Make sure you have a backup of your phone FROM iOS 6! This process will reset your phone to factory before restoring it, and you won't be able to restore it using an iOS 7 backup! You will also need to know the Apple ID and password for the phone's primary account, your backup password (if you set one - see if "Encrypt backup" is checked on your phone's "Summary" page.) DO NOT PROCEED IF YOU DON'T HAVE ALL THREE OF THOSE THINGS!
1. Download the old iOS version you need - this is a fairly large file, about 1 GB in general. If you've got a GSM (AT&T or T-Mobile in the US) iPhone 5, you can get it from http://appldnld.apple.com/iOS6.1/091-3403.20130502.Xerz1/iPhone5,1_6.1.4_10B350_ Restore.ipsw and otherwise, you can find it on that same site, but you'll have to look around. 5's can run 6.1.4, 4s's and older need 6.1.3. Note that there are different versions of iOS for GSM and CDMA (Verizon or Sprint in the US) networks, so make sure you get the right one. Save this file somewhere safe.
2*. Put your phone in DFU mode. http://theiphonewiki.com/wiki/DFU_Mode contains instructions to do that. Note that you must already have iTunes open to do this.
3. When iTunes notices your phone in Restore mode, click "OK" and it should pull up the restore page for the phone. Press Alt (or Option for Mac) and click on "Restore iPhone..." You must press Alt/Option, or this part won't work.
4. That will open an Explorer/Finder window. Point that at the old iOS .ipsw file you downloaded in step 1. That will restore the iOS 6 software, which will take a few minutes, typically.
5. Once iOS is restored, iTunes will prompt you for the primary's AppleID and Password. This is to prevent theft. Enter them.
6. You can then restore your phone from a backup or set it up as a new phone. I made a local backup before I installed iOS 7, so I restored from that - hopefully you did the same, but anyway, you're best served restoring to your newest iOS 6 backup. Note that this will not allow you to restore an iOS 7 backup! Obviously, if you have a backup password, you'll need to enter it here.
7. Your phone will auto-sync. You can stop that if you want and sync it later, when you've added your podcasts back, or you can let it finish if you want.
8. You now have a phone running iOS 6 again. You can join me in downgrading back to iTunes 11.0.5 and waiting for Apple to fix this show-stopping podcast bug.
*Step 2 is necessary if your "Restore iPhone..." button is not present or grayed-out, as mine was. If it's not the case for you - if your "Restore iPhone..." button is there and operable, you can skip step 2 and procede to step 3 by Alt-clicking on that "Restore iPhone..." button.