The built in podcast app checks on podcasts only when it determines is a good time to. It checks more often when plugged in and on wi-fi. If you want to "pull" a podcast, there are several podcast apps that get good reviews in the store. One that I use to pull a podcast is bringcast. It has a free trial.
As an additional note, with the addition of the new ...
Open podcasts app.
Open in-app settings.
Choose how much episodes you want to keep.
The older ones will be auto deleted.
Currently there is no option to delete podcasts once you have listened, in the built-in podcast app.
The built-in Windows Phone 8.1 Podcasts app supports this.
Beside the podcast artwork display, there is a grey "1.5x" label. Press this once to select it (it goes white) and playback runs at 1.5x speed; press it again and it switches to 2x speed. Press it once more to deactivate it (switches back to grey "1.5x").
There is a new app that solves the problem. You download podcasts to your computer using any software (like Zune) and use the new Windows desktop application for sending them to the phone. You can get the new application at http://www.windowsphone.com/en-us/how-to/wp8/windows-phone-app-for-desktop
Resolved it, and I'll post the answer here but it's probably unusual to run into this problem (of multiple @live accounts) so I don't know how useful it'll be.
I think the problem was that the phone was associated with a newer @live account and on the zune/desktop my default account (and all podcasts) were setup with an older @live account. It seems as if ...
In oder to get podcasts into your podcast app you actually have to subscribe to one, either by searching our by entering a feed.
You still can listen to the mp3s though the music app.
It might be easier though to try one of the audio book apps from the store - I think those offer the functionality you're looking for.
To subscribe to a podcast in windows phone 8.1, you either enter the url to the podcast feed, or search inside the app. The app uses Bing to find podcasts, not the store any longer. Currently, these are the only two ways to subscribe to podcasts. The phone then manages the listened to episodes.
In regards to the lack of support on WP8, I have to chalk it up as a failure in planning. The official word from Windows Phone Support is that there's no way to get files that aren't from the store in to the podcast section.
If you really want this feature, for now all we can do is throw three votes at it on the official feature request site.