Overriding theme_menu_item

Hi, everyone.

I spent the better part of the last 2 days searching for a way to do that - add a "level-1", "level-2" etc. class to the menu's <li> elements. Anyone done that? I really need to do it and can't find a way. There is a snippet that does the same thing for the links inside the <li> element, but that's not what I need. I need to add the class to the <li> itself. So, what i need is:

<ul class="menu">
    <li class="level-1">Level 1</li>
    <li class="level-1">
        <ul class="menu">
            <li class="level-2">Level 2</li>
        </ul>
    </li>
</ul>

I hope you can help,

Bitstream.