Transitions on the CSS display property

I'm currently designing a CSS 'mega dropdown' menu - basically a regular CSS-only dropdown menu, but one that contains different types of content.

At the moment, it appears that CSS 3 transitions don't apply to the 'display' property, i.e., you can't do any sort of transition from display: none to display: block (or any combination).

Is there a way for the second-tier menu from the above example to 'fade in' when someone hovers over one of the top level menu items?

I'm aware that you can use transitions on the visibility: property, but I can't think of a way to use that effectively.

I've also tried using height, but that just failed miserably.

I'm also aware that it's trivial to achieve this using JavaScript, but I wanted to challenge myself to use just CSS, and I think I'm coming up a little short.