1pixelout.net made a great free mp3 flash player; they made it mainly as a wordpress plugin but it is very easy to implement it on it's own.

To implement it you only need 3 files:

audio-player.js
player.swf
an mp3 file

You can embed it like this:

<script src="http://www.yoursite.com/audio-player.js"></script>
<object width="290" height="24" data="http://www.yoursite.com/player.swf" type="application/x-shockwave-flash">
<param name="id" value="audioplayer1" />
<param name="FlashVars" value="playerID=1&amp;soundFile=http://www.yoursite.com/some_mp3.mp3" />
<param name="quality" value="high" />
<param name="menu" value="false" />
<param name="wmode" value="transparent" />
<param name="src" value="http://www.yoursite.com/player.swf" />

</object>

Replace the bolded code with your own paths.

Example:

You can find more settings here.
You can find another tutorial about embeding it here.