Ben,
Great question. The maximum file upload size inside of WordPress is actually set by your web host. Some hosts set the limit as low as 2 mb, other shoot for 20.
I’d start by asking your web host if they will increase that for you. Some have things set so it can’t be changed, but others will increase it once you ask. That’s the easiest solution.
The other option is to use a FTP program to upload the file to your site. WordPress stores uploaded files in a folder called wp-content/uploads/ where you could upload your podcast files. You could create a folder inside it called podcasts to keep them organized.
By uploading the files with a FTP program, you can skip the maximum upload file size. All you’ll need to do is enter the URL to the podcast file into the blank. The URL would follow this format:
[audio src="https://www.yoursite.com/wp-content/uploads/podcast/yourfile.mp3" /]
You’d just replace the http://www.yoursite.com with your site’s address and the yourfile.mp3 with the name of your file.
If you go the FTP way, remember that you can’t have any spaces in your file name in order for it to work.
Let me know if you have any trouble,
Bill