I am having similar issues. Several podcasts work fine, no problems. For others, I get persistent problems.
A few I can't download or stream at all. Some I can stream fine, but when I try to download, it stalls at "Preparing to download..." forever. Sometimes this eventually fails with a notice that the podcast couldn't be downloaded, other times, it just stays that way indefinitely. A few times by restarting the phone (not just the app, the whole phone), a podcast episode that was stalled before will start downloading. Its inconsistent and frustrating. None of these issues seem to occur in iTunes on the Mac, also with the same internet connection.