I
believe—don't quote me on this, I haven't lived in a shoji-panel house either—you can't really get a "breeze" going through a shoji-panel house, in the sense of air
pushing its way through the house (without opening the doors on the intake side and so negating the air-cleaning benefits); but what you
can do is get air to
pull through the house, which is sorta
like a breeze. It's fresh air, rather than stagnant/humid air, but it won't dry your laundry.
I believe how this works, is that you wait for wind or sun to be hitting the house on one side, and then open the windows/doors on the opposite side of the house from the wind/sun. Then, either the vacuum created by the turbulence at the far side of the house (in the case of wind)—or the air-pressure difference between the hot and cool sides of the house (in the case of sun)—will create a low-air-pressure area on the outtake side, and so suck air out of the house through the outtake gap. There's then a pressure difference between the inside of the house and the high-air-pressure on the intake side; and since shoji is permeable, the house will suck air in through the shoji to balance the pressure difference.
Probably, in a modern house designed to take advantage of the same principle, you could avoid having to run around opening and closing doors, and instead passively automate this, by putting low-friction one-way valved vents on all exterior walls of the house. Then, no matter which way the air-pressure-gradient formed, the house would release air in some direction and then intake air from every other direction to fill it.
I believe this system falls down when the house is equally-lit (e.g. at noon) and the wind is stagnant (e.g. when you're in a high-pressure system.) But at that point, there's also no wind to blow dust in, so you may as well open all the doors and turn on a fan.
Also, rather than needing a passive environmental pressure difference, you could just run a blower fan that pushes air out of the house, e.g. an oven hood fan or bathroom vent fan. That'd also create an air-pressure difference with the power to suck air in through the shoji.