Overcast Radio, LLC has Overcast: Podcast Player (iTunes Link) for the iPhone, iPod Touch, iPad in the App Store for FREE (was $4.99 via in-app purchase). Recently updated versions 2.0 unlocks all features, which was previously a $4.99 in-app purchase. The app is now supported by voluntary patronage. This is a great app, if you can, I would suggest supporting continuing development of the app.

A powerful yet simple audio podcast player, with features such as Smart Speed, Voice Boost, and Smarter Playlists to help you listen to more podcasts in more places, try new shows, and completely control your experience.

Overcast is a modern, fully featured audio podcast player with many useful features in a simple, intuitive interface: