OK, I found a solution here. So what you need to do is include the WMODE parameter:
<param name="wmode" value="transparent">
You should be able to include that line somewhere in the code for the flash player, as long as it’s inside the <object> … </object> tags. I’ve added it to my sandbox site, so now you should see the movie slide under the menu in IE.
Here’s my code for reference:
<object classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="https://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,29,0" width="400" height="300">
<param name="movie" value="/video/tangramone.swf">
<param name="quality" value="high">
<param name="wmode" value="transparent">
<embed src="/video/tangramone.swf" quality="high" pluginspage="https://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" width="400" height="300">
</object>