Question
I have a piece of land about 300ft by 1000ft.  The water table is anywhere from 0 to 12", so there is a lot of surface water after a heavy rain. Most of the land is wooded.

My question is this-if I dig a trench/stream running the length of the land and emptying into low natural drainage area, will that solve the soggy land problem?
Thanks.

Answer
Maybe.  It depends a lot on the soil structure and the grade of the plot.  You will undoubtably get some drainage with a single trench, but if your goal is to throughly drain the area you may have to do something like a grid with a main trench and some side trenches feeding into it.  
One thing to consider, if the trees on the plot are native and are adapted to the conditions, changing those conditions may have an adverse effect on the trees.