Thanks for all the feedback - please keep posting your constructive comments here, as this is a huge help for us in gauging community needs. The designers are continuing to evolve the embedded player to address some of the concerns raised here while still providing new ways for people to discover videos. You may have already noticed the following changes: related videos only show when the menu button is pressed, mouseover function is no more, and the menu for related videos now displays at the end of the video you've just watched.
If you'd like to turn off the option to see related videos through your embedded player, simply add "&rel=0" after the video ID number in the embed code. Here's an example of what your embed code should look like if you'd like to turn off this feature:
type="application/x-shockwave-flash" wmode="transparent" width="425"