Can I still make a playlist of podcasts with the new podcast app?

No, for some reason it's not included as one of the features of the Podcast App.

If you need playlists (as many of us do), then delete the Podcast App and podcasts will return to the music app, complete with podcast playlist functionality.

After deleting the app, I had to turn off my iPhone and restart it again. Before that the playlists were still gone.