For anyone looking for a solution to the original problem or one like it, the answer has already been given by DOWNSIDEUP:
So, for example, if you open a movie in VLC and drag it over to your secondary monitor or (as in the case of the original post) you open an MLB.tv game and drag it over, the translucent menu bar will *always* show on the secondary monitor. The external monitor is a separate space (unless you've deselected that option in System Preferences, which is kind of a silly thing to do), so if you drag a window over to the second screen, the application is actually running out of your machine's space and not from the secondary monitor's space. So in the eyes of the external monitor, the focus is still on your machine's space when you're running a dragged VLC or browser window in full screen on the secondary monitor. So it displays that obnoxious translucent menu bar.
So long story short (too late!), literally all you need to do is close the application, put the focus on your secondary monitor's space (by switching to an app running in that space or using the dock on that space or whatever), and then reopen the application from within the secondary monitor's space. Now when you go fullscreen, the menu bar won't be there, because the focus of the application is in the space it's running fullscreen in.
(Side note: When Scott says, "Okay, Safari will show full screen video without the menu bar, Chrome Canary will not", I guarantee he opened Safari from within the secondary monitor's space, but the Chrome process was the same one he initially started on his machine's space. I would bet the farm that if he quit Chrome and reopened it from within the secondary monitor, it would display the MLB.tv game without the menu bar.)