Hi Michael,
I was doing some testing and everything works fine for me.
There is setup I have used.
1. I have created audio folder in root of my website application and put files player.swf and audio-player.js into it.
2. I have add file test.mp3 into this folder too (it was part of audio player package but it contains no sound, so nothing could be heard, in fact it was little bit confusing)
3. I have added this html text into static html webpart in source mode:

<script language="JavaScript" src="/KenticoCMS_3.1a1/audio/audio-player.js"></script>
<object type="application/x-shockwave-flash" data="/KenticoCMS_3.1a1/audio/player.swf" id="audioplayer1" height="24" width="290">
<param name="movie" value="/KenticoCMS_3.1a1/audio/player.swf" />
<param name="FlashVars" value="playerID=1&soundFile=/KenticoCMS_3.1a1/audio/test.mp3" />
<param name="quality" value="high" />
<param name="menu" value="false" />
<param name="wmode" value="transparent" />
</object>

Please note that /KenticoCMS_3.1a1/ is path to my website folder under the root of web and it should be change according to your folder structure.

4. Everything was working correctly

Best Regards,
Zbysek Nemec