Your question is an interesting one. In the outdoors, temperature gradient (change with elevation) and to a lesser extend change in wind speed with elevation, are known to affect sound propagation. That is an effect called refraction. Cross wind should also affect the propagation of sound, "sweeping it downstream" as it were. It is exactly like making waves on a river. The waves are swept downstream along with the flow of the water. If you are floating down the river on a raft, the waves appear to spread equally in all directions. Seen from the shore however, the waves are seen to spread more downstream than upstream. This can be seen clearly on a fast-flowing river because the speed of the waves are comparable to the speed of the river flow.
The problem comes with measuring this effect for sound waves, because sound waves propagate so much faster than a house fan airflow. The speed of sound in air is about 330 meters per second, whereas I guess that windspeed from a fan would be at most 3 meters per second. The effect will be very small, of the order of 1%. It is going to be a difficult effect to measure, but not necessarily impossible to measure. I would not bet against it being possible, but I have not heard of it being done. This is why this is quite a good experiment to conduct: the answer is not known in advance!
My advice to improve the chances of success would be:
- use a high frequency for the sound, preferably above 3000 Hz. High frequencies scatter around the room less than low frequencies. At low frequencies you will be measuring more the effect of the echo than the direct sound from the loudspeakers.
- use time differences rather than intensity differences. You can measure minute differences in timing between the microphones by averaging over long periods (several seconds at least) better than you can measure absolute amplitude changes before and after you turn the fans on.