Everyone else has been speculating, so I thought I'd throw my hat in the ring.
Briggs wants out of Chicago. Washington wants him. KC needs to ship Green and Chicago needs a QB. So why not the following:
Bears trade Briggs to Chiefs for Trent Green and Chiefs #1 pick(23)
Chiefs send Briggs to Washington, Chiefs receive Washington's #1 pick(6).
Bears get their starting QB
Washington only spends a 1st rounder on Briggs
And Chiefs move up to #6 pick in 1st round and cut some salary cap.
Everyone wins. The Chiefs would then have a shot at getting someone like Gaines Adams(DE) or Joe Thomas(OT).