Really interesting thread.
Personally, I like the sit and stare method. Have never tried the cardboard cut out thing yet. There was a WWII aircraft engineer who would cut out pieces in scrap, fit them to a plane and then tell workers to "do that"
I use this almost daily with a twist;
"POWER NAP. What is it about genius and a disdain for sleep? Thomas Edison was another who believed most people sleep far too much and are unproductive as a result. Thomas Edison was a fan of the power nap. He gave it a good twist, though, which he claimed was integral to some of his best ideas. Edison would sleep sitting upright in his chair, elbow propped on the arm with a handful of marbles. He would think about his problem until he fell asleep, and soon enough he would drop the marbles on the floor. When the racket woke him up, Edison wrote down whatever was in his head, regardless of what it was—creative solutions, new ideas, a reminder to pick up milk on the way home."
Instead of sitting in a chair, when I first wake up in the morning or becoming fully awake, I try to think about what ever puzzle I am working on. The mind is most creative when some where between asleep and awake. I keep a notepad and very often parts near by.
But, say with MBs. I find the best way to get those little "ah-ha!" moments is to just stop, relax and enjoy pondering them. Just sitting by a bike and staring at it with a cup of coffee or what ever leads to those moments.
Recently I was shingling the roof on the shop. Stopped for a moment and had one of those moments. That set into motion about 6 months of work and large expenditure. Might end up being a really cool endeavor.