WORDPRESS: Display list of Sub-Categories and the posts they contain, within one main Category

Question

I've found tons of code and plugins to do various things; from show posts for specific cats, subcats of a cat, etc.. BUT, I cannot for the life of me find, nor do I know the WP API well enough to do what I need with it..

Here is what I'm trying to accomplish:

Display a UL of all subcats within Cat31, and the posts for each of those subcats:

SubCat1 Post 1 Post 2

SubCat2 Post 1 Post 2

SubCat3 Post 1 Post 2

It's pretty straight forward, but all the loops I have tried fail either at the subcat loop or the post loop (one or the other works, I cannot get them both to work..)

So, unless I can find a plugin to do this (I'd prefer to code this into a template file!) then I need to figure out how to:

Loop Subcats within Cat31 while looping subcasts, loop posts for each subcat

Any help is GREATLY appreciated!