I know there's another similar thread about this, but it's been resolved for the original poster.
I'm having a similar problem, where only the "Download" link shows up for any episode. I just installed a brand new copy of WordPress and Seriously Simple Podcasting today.
The MP3/OGG/WAV files I'm using have been uploaded via FTP to a subdirectory of the domain. I included them in the episode posts by linking to their full URL. Their size/duration is correctly pulled from them.
Here is an example entry: http://doomradio.org/podcast/interview-with-xaser/
MediaElement.js never seems to get included anywhere in the page, so it's not a surprise that the player never shows up. I have tried with the default Twenty Twelve theme, Twenty Eleven, and another different theme.
Am I doing something wrong?