Jan, I use a laminate floor cleaner because that's what I have throughout, but you could use a floor cleaning agent that doesn't require rinsing. Doesn't Lysol, that pine cleaner or Mr. Clean have one? I'd mix a weak solution in a spray bottle and just SPRAY as you go long so your floor doesn't get too wet.

To avoid streaks, I go ahead and rinse my floor since I have soft water. You'll have to see what you think of your floors once you pick a cleaner.