I like pod.ding (currently being updated and not available in the store). It allows you to do what you ask, but to download the podcasts in the morning you have to open the app, then it automatically looks for and downloads episodes. The developer is also very receptive to feedback and suggestions.
Right now I am using ipodcast, which works well, but isn't as nice to look at. One nice thing it has is the ability to backup your podcasts to the cloud.
The reason I don't use the built in app is it is difficult to keep many episodes of a particular podcast. It has some quirks with the settings that if you tell it to keep the most recent X podcasts and then delete them and mark as played, the app will automatically try to download the X most recent not listened to episodes. So basically it will always keep X episodes on the device at all times, or maybe that is just my phone, but it was very annoying.