The three common ways to measure wind speed are :

1. Hot wire
2. Rotating cups or fan
3. Vane

The only one that doesn't depend on wind direction is rotating cups.
A hot wire doesn't have to be directional, but most are.
A vane definitely has to turn with the wind.

I once made rotating cups from old LEGGS pantyhose containers. They are plastic and egg-shaped. When opened, they are fairly hemispherical plastic. I epoxied 3 of them to stainless steel rods about 15 inches long. Those rods were press-fitted into a machined aluminum cylinder that was attached to an old 5.25" disk-drive motor. The motor provided the bearings and the speed indicator. The motor had a separate, three-wire (amplified hall-effect) tachometer that didn't require powering the motor to provide an output of 5V P-P.
It worked for awhile, but the epoxy and plastic LEGGS containers couldn't withstand the elements. The sun's ultraviolet rays made the plastic brittle, and the rain and the ultraviolet rays made the epoxy soft. You could probably devise a better way. By the way. I calibrated this device by sticking it out a car window and driving down a road at various speeds, and then down the same road in the opposite direction at various speeds and averaging the equivalents.

Another approach that I used (this one worked) was using a PTC (Positive Temperature Coefficient) thermistor. I chose one that had a knee temperature of about 70C and would reach that temperature when powered by 12V. I put a small (10 ohm) resistor in series and boosted the voltage drop across that resistor with an op amp.
A PTC thermistor is like a self-regulating heater. If you apply sufficient voltage, it will heat up (due to its resistance) up to the knee temperature. At that point, the resistance goes up rapidly. The increasing resistance lowers the current through the device and it cools back down again. In other words, it will maintain a given temperature all by itself by regulating its own current. If you blow air across it, the air will cool it down. The PTC thermistor will immediately sense this and lower its resistance in order to maintain its temperature (at the knee value). If you measure the current draw of the thermistor, you will get a reading that is dependent on the cooling effect of the air - which depends on two factors: #1. The ambient air temperature and #2. THE VELOCITY OF THE AIR. The cooling effect of air is proportional to a constant times plus the square root of the velocity. So, if your PIC knows the ambient temperature and the current through the PTC, it can calculate the wind velocity. PTCs are generally disk shaped, however, so they are somewhat wind-direction dependent.

I always thought a vane would work but never built one. I was going to hang a piece of metal from a low-friction bearing and measure the angle of deflection.
using an optical method.