The best way to compare two items for their "greeness" is to look at several factors: Environmental Impact, Operating Cost, Energy Use, Manufacturing and Cooling Effectiveness.
Environmental Impact - It's not hard to say that the fan wins this category. Air conditioners use refrigerants called fluorocarbons. Even though the newer generation of refrigerants (Puron) are better for the atmosphere the creation of these products are much more energy demanding than any components of a fan. Just manufacturing an air conditioner creates significant environmental pollution compared to the fan.
Operating Cost - Fans win this category also. Running a fan is significantly cheaper than running an air conditioner. Air conditioners also require yearly maintenance which adds to the operating costs.
Energy Use - It is estimated that a typical ceiling fan uses 1/30th the energy that a standard window air conditioner uses. Therefore you could have 30 fans throughout your house blowing on your every position for the cost of running one window air conditioner in a room that you probably only occupy for a small portion of the day. Again, the fan wins big in the category. You can also use a whole house fan which cools down the entire house for a fan system that's even more efficient.
Manufacturing - Fans win this category as well. As I pointed out in the environmental impact section the manufacturing process for air conditioners is much more involved than fans. Not only is the unit more demanding but the refrigerants are as well. Fans are very simple machines that use a small motor and a bunch of simple components. Air conditioners have many complicated working parts including a compressor, an evaporator, coils, filters and gas compounds.
Cooling Effectiveness - Obviously the air conditioner wins this category especially on those really hot, humid days when it's even hard to breathe. However, if you eliminate the really bad days the fan still does a very effective job at cooling a person down. One of the most important factors in how we feel cool is the movement of air over our skin. So an air temperature can be warmer but be flowing over our skin and feel just as comfortable as a room chilled down with no air movement.
Conclusion - Obviously if you're talking about which cooling device is greener the fan wins hands down. Strategically placed ceiling fans, exhaust fans and window placements can provide a very comfortable living atmosphere even in warm climates. Choosing fans over air conditioners is certainly the green choice. If you don't have ceiling fans in your existing home and you'd like to install some in place of your existing light fixtures then visit my other blog and read How To Replace Light Fixture with Ceiling Fan.