Today, Amazon announced that you can now use Apple Podcasts and Spotify podcasts with Alexa. All you'll need to do to get started is enable it through the Alexa app, and link your Apple or Spotify account. What's even better is that once your account is linked, your place in the podcast is synced. That means that you can start listening at home on your Echo, and then pick up where you left off from your phone in the car.
You'll just have to say, "Alexa, play Windows Weekly on Apple Podcasts", or you can even customize it by saying to play last week's episode or something like that. You can tell Alexa to play the next episode, to fast forward or rewind a certain amount of time, and so on.
Naturally, you'll be able to set either one as your default podcasts app. You can do this by going to Settings -> Music & Podcasts -> link/manage new services.
Right now, the feature is only available on Amazon's own Echo devices, and Sonos. Hopefully, it will come to more third-party devices over time.