Why am I not burned by a strong wind?

Air molecules $(\require{mhchem}\ce{N2_}$ and $\ce{O_2})$ have an average speed of around $500\text{ m/s}$, varying some depending on the temperature. This means that a nice $5\text{ m/s}$ wind is a hundred times slower, and the energy represented by wind is 10,000 times smaller than the thermal energy. Therefore, wind does not have considerably more energy than calm air and will not burn you.

Very high-speed winds, such as those in tornadoes, hurricanes, or the wind you would experience while sky-diving, are still only around $50\text{ m/s}$, so the energy density in the wind is still just 1% of the thermal energy density. Likewise, the ram pressure the air exerts on you would be small compared to the homogenous atmospheric pressure, so no large effects should be observed. Thus, one would not expect even high winds to burn you.

The transfer of heat between you and the air is fairly complicated, and does not depend solely on the energy density of the air. Wind usually makes you feel colder, in fact. Heat travels across gradients of temperature. The air right next to your skin will be at the same temperature as your skin, but the air a small distance away will be at the ambient temperature. This creates a gradient of temperature, and heat travels across the gradient. When there is wind, the difference in temperature between your skin and the ambient air is the same, but the temperature falls down to the ambient temperature a shorter distance from your skin. This increases the temperature gradient, so that you cool down faster with a wind.

Humidity also plays a role; heat transfer is not very simple. However, I think this suffices to explain why we should not expect wind to burn you. You will burn up if you travel through the air at extremely-high velocity. This happens to meteors and other astronomical objects moving at orbital velocities ($\sim10^4\text{ m/s}$) when they enter Earth's atmosphere. It is also relevant for fast-moving aircraft, which do experience winds as fast as the thermal velocities of the molecules in the air. I've heard it said that the SR-71 Blackbird, the fastest airplane ever built, heated up so much due to aerodynamic heating that it had to be built to be loose at low speed so that the parts would fit together at top speed. See "Aerodynamic heating" for more.

The other answers address your question quite well. Just as a reminder of the ability to be burned by a strong enough wind, the image below shows the Chelyabinsk meteor during entry into Earth's atmosphere last year over Russia. :)

If heat I feel is just lots of particles going wild and transferring their energy to other bodies, why am I not burned by the wind?

I think the most direct answer to your question is that heat is the random movement of molecules, with speeds on the order of $v_{rms} = \sqrt{\frac{3RT}{m}}$ which is in the hundreds of meters/second range, whereas the layer of air surrounding you moves in a not-quite random way (there is a boundary layer, for example), and not usually as fast.

If we think of friction, on the other hand, I think the problem is that while friction /drag will heat you up and the air surrounding you, the stream of air is also very efficient at removing heat from you, so depending on the temperature of the wind, the two effects will partially cancel each other out.

Now, when you start getting to transonic speeds, the (adiabatic?) compression of the air in front of you will heat it up significantly (I believe this is why the SR-71 was made out of titanium). At hypersonic speeds, this can give you a nasty case of plasmification.