Can we have air flow without air noise? I want to have air flow in my room, but I don't want a fan to move the air because of the air noise. I think the noise comes from all air molecules going to the same direction. If I can move the air molecules in many different directions, could I create an air flow without noise?
However, if the air molecules move in many directions, wouldn't the room temperature increase? I want to make myself cooler not hotter.
 A: The noise from a fan is mostly the vibrations caused by the fan motor, and the rush of air past the fan blades.  
You can buy a quieter fan.  It will cost more.  Some new fans have no moving parts, no blades :
http://electronics.howstuffworks.com/gadgets/home/dyson-bladeless-fan.htm
The obvious easy way to create an air flow (cross-breeze) without creating noise is to open a window and a door.
If you keep a fan running the room will gradually become warmer, mainly because much of the electrical energy is dissipated as heat in the fan. 
A: I will address your question on how the fan actually works and cools you. @sammy gerbil answered the other part about the noise. I can only add that you probably confuse the fan noise with the whistling in the ears when the wind is strong.

Fan creates a flow of air — i.e., increases the speed of the particles in particular direction. Naively, this would mean the increase in temperature.
But this is an ordered flow on a scale much larger and with speeds much lower than a typical particle speed (for scale: the speed of sound in air is about $300 m/s$ — more or less of the order of a typical molecule speed). So this flow is barely noticeable by individual molecule. 
You can split the mass of air into pieces (small in terms of the flow but large in terms of air molecules), that are roughly comoving. In a frame of reference where a particular piece is stationary, the air won't be different from air in a room without a fan.
It turns out, the fan does not influence the temperature of the air directly (except for heat losses mentioned by sammy gerbil and heating through dispersion of flow energy). Different effects are important. 
1. Mixing the air
For starters, consider a human in the room. Human is a power source that constantly produces heat (although at a small rate), so the air around the human will be slightly warmer than in other places. If you force this warmed-up air to move somewhere else, the human will feel a bit cooler.
2. Moving the water vapor
Sweating is a natural mechanism of the thermoregulation. When we sweat, we form a slim waterskin with large surface. Although in general water molecules stay together, the most energetic ones are able to break the bonds and fly away — and the more surface you have, the more molecules escape. When the most energetic molecules are gone, the total energy of the water is smaller — hence it becomes cooler.
However, if the air is filled with water vapor (for example, the same escaped molecules), evaporation becomes inefficient because while some molecules get loose, others come back. This is the place where the fan helps us: again, it moves the moist air away allowing us to evaporate the sweat.
But why fast molecules do not mix themselves? It turns out that molecules never travel far in a straight trajectory. They often collide and change direction. For example, the mean free path in the air is about $68 nm$ according to Wikipedia. So it would take really long for a particular molecule to cross the room by itself.

Regarding your idea of a flow in many directions: I think you are basically going to put more but less powerful fans in the room facing away from each other. That is probably inefficient because of the heat losses and inability to create a steady flow around the room that will circulate the vapor and hot gas.
A: How about piping the fan in from another room or basement, thus keeping the fans out of ear reach.  There are plastic vent hoses that are easy to manipulate.
There are thermo-electric coolers but they are horrible inefficient, but silent.
A: Even if you could silently create air flow in multiple directions you would not increase cooling. Temperature is a measure of the random kinetic energy. You would introduce random kinetic energy - not flow. You would actually increase the temperature of the air.  
Even introducing uniform flow it would eventually convert to random kinetic energy and increase temperature.  If you do an energy balance the fan has done work on the air - them temp has to increase.  
Sound is a percussion wave and truly random motion would not be a percussion wave.  Mechanically produce random motion would not be realistic. It would need to be on a molecule by molecule basis.  It would be at the level of a Maxwell speed daemon.  You could install heat radiator and silently introduce increased random motion but that is definitely not going to cool the room.
