SWF that is linked to an mp3 (like you have now) is in a way wrapped using SWF. It plays because the SWF player is a plugin that knows what to do with the .mp3 MIME type. If you don't use SWF then not all browsers support .mp3 (like Firefox) so you can't play them.
HTML5 <audio> does today support mp3; however not all browsers support it, so the browsers are the limiting factor.
As of today Oct 03, 2012 Firefox does not natively play an MP3 file due to patent licences from a number of organisations (Technicolor/Thomson Consumer Electronics, the Fraunhofer Institute, Alcatel-Lucent, and Sisvel). In March Mozilla announced that they would support html 5 and mp3 native formats but as of now it has not yet been integrated into Firefox. Mozilla is trying to keep their browser open-source and it is harder to do when supporting such proprietary formats. Mozillia appears to plan to put it in Droid. So in Firefox (and other Mozillia browsers) you still need to fall back to a SWF audio wrapped file.
Well there are two parts to this.[INDENT]1) Your web server needs to have the proper mime-types associated with your site. If you use Apache as your webserver you can define these MIME types in a .htaccess file like:
<font color='#880000'><font face='andale mono'># AddType TYPE/SUBTYPE EXTENSION
</font></font><font color='#660066'><font face='andale mono'>AddType</font></font><font color='#000000'><font face='andale mono'> audio</font></font><font color='#666600'><font face='andale mono'>/</font></font><font color='#000000'><font face='andale mono'>mpeg mp3
</font></font><font color='#000000'><font face='andale mono'><font color='#660066'>AddType audio/ogg</font> ogg
[INDENT]2) Your browser needs to support mp3. As mentioned above Firefox doesn't support it yet, but supposedly will in the future.
It is because of this - not all peoples browsers support the same MIME audio types - that html5 audio should not be served with just one audio type. The HTML5 spec allows the browser to detect if it supports one format and if not then will try the next format. I do however understand that you are exclusively using .mp3, so I'm not sure that HTML 5 audio does make sense for you; however it has many advantage so it is a shame.
For more info go to the Mozillia developer network