WP seamlessly support media streaming content hence using podcast in wordpress is easy, you will need to just link to an audio file with your post and WordPress will automatically add the essential enclosure tag to your RSS2 feed to make it accessible throughout the web.

Important: Use a absolute URL when linking to the audio file. Otherwise WordPress will not make an enclosure for it.
Wrong: My podcast

Right: My podcast
That’s it — just link to an audio file and WordPress will take care of everything for you.
Check another step; You need to make your RSS2 feed available. Be sure to put a link somewhere on your web page:
http://example.com/wordpress/?feed=rss2
Or, if you’re using Atom:
http://example.com/wordpress/?feed=atom
As of WordPress 1.5, the RSS 2.0/Atom links are located on the bottom of the page by default.
To make life easier for iTunes users, consider also linking to your podcast feed with the itpc:// protocol rather than http, e.g.:
itpc://domain.org/path/?feed=atom
iTunes and other such services allow listeners to subscribe to your RSS/Atom feed and automatically download your content to their portable audio players as it becomes available for your users. However make sure that the linked file is accessible when you publish your post otherwise, WordPress would not be able to gather the information it needs from the linked files. If you change a linked file, republish the post which will enable WP to update its file size information about the file. Also for better coverage, you may require to put .xml at the end of the RSS/Atom url, to let it work flawlessly.