What's my brain doing when I'm not looking?

What's the best way of problem solving? "I'll think about it" you say when confronted with a tricky problem - but it appears this is probably the worse thing to do.
There's an interesting extract in Saturday's Guardian (here) about how the different halves of the brain deal with information differently - left side is logical, right side is doing something much vaguer. Some neuroscientists are saying that this vague stuff the right side of the brain is doing is actually really very important in problem solving.

What the inside of my head feels like most of the time

When I'm stuck with a problem (which is usually in programming, maths or engineering - it's what I do), I do tend to sit down and think about it. Then stand up and think about it. Then get frustrated after an hour and go and do something else - and it's then that the solution often comes. And this is without trying - so there must be something going on in my head, which I'm not conscious of, that's making connections. I think this is the right side of my brain at work.
But sometimes it's slow - I still get small moments of revelation when something I learned at college suddenly makes sense - and I graduated almost 20 years ago.
This also explains why it's difficult to teach problem solving. You can go about it logically - but actually it's the weird connections the right side of the brain is making that finds the solution.
So "I'll sleep on it" may be a better approach when I've got a problem to solve. I wonder if I could get away with invoicing for time spent asleep...