I think that the "as part of normal movement" should be taken basically as part of a normal movement action.
Now, some people think you can do it during a charge, but during a run seems to be stretching it a bit, after all you can't do anything else than move in a straight line, and you even incur attacks of opportunity.
So, I'd say definitely no, you can't tumble when running.
You'd have to take a double move away first, and then run. If all you do is move away, with a single or double move, in a round, you don't incur an AoO (as long as you'r not surrounded or somesuch).