Because riding a motorcycle, I didn't feel a difference when riding in different directions.
To really test the hypothesis, you should ride a balloon: it has no connection to the ground, there is no effect that will make it go slower or faster than the wind. I hear it is a really calm experience. I guess you hear something only if the wind direction changes, i.e. turbulence near the ground or thunderstorms (you do not want to be near one in a balloon), or wind direction change at the altitude you are in, or reaching an altitude with a different wind direction.
Hypothetically, if you were on a flat plain with constant wind speed, and riding at wind speed, the only noises you'd hear are road noise from the chain, tires, gears and engine.
Normally, however, wind speeds are far lower than the motorcycle speeds where you start to really notice wind (less than 15 mph on average even in (most) really windy cities). And when you start getting 30+ mph winds, they tend to be rather gusty instead of constant. Also, high speed wind tends to run into nearby objects like trees, telephone poles, and even curbs, all of which makes wind noise.
I don't ride motorcycles, but I know in a car you can hear and feel the difference when riding relatively closely behind a semi trailer. It won't go away completely, but the semi is sucking a lot of the air forward, speeding it up, so the difference between you and the air around you is lower.
You can easily see the difference in air braking too. Come up behind a 50 mph semi at 70 mph on a multi-lane road. Let off the throttle completely, then change between the lane behind the semi and the other lane. You'll slow down a lot faster when you're in the other lane as a result of the pressure difference. You should be able to tangibly feel the difference on a motorcycle, since the wind hits you harder in the raw air. Or even just sticking your hand out the window.
You can also see this in fuel economy. Drive into the wind and you'll get worse economy than driving with the wind to your back. I once had 18 mpg one way and 29 mpg the other on a Dodge 2500 turbodiesel truck just west of El Paso, Texas. Likewise, you can get better fuel economy by riding behind a semi, although you need to stay far enough back that you're not constantly adjusting the throttle (though generally you're plenty far for that just by virtue of leaving a safe reaction distance between you).
If your ears moves continuously as per the wind the relative speed will be zero , and you wont hear it. But this case is ideal but not practical.
In reality, your movements wont be as smooth as wing. so there will be a relative difference (Non Zero) between both the vectors, your movement and wind.
Therefore, you will be able to feel it and hear it.
Yes you would and no, you never do.
You never hear the wind per se, you hear only its interactions with physical objects. If you're moving in perfect unison with it, it is not interacting with you and so makes no noise from you. You will, however, hear it interact with other physical objects around you. (E.g. Trees, traffic cops, acorns left on the road by squirrels, etc...)
In the vacuum of space, you would not hear the space wind if your space bike was moving in unison with it. Unless, of course, a space squirrel left a space acorn near your path.
When sailing against the wind, you feel it as much stronger because the true wind (wind you'd feel when standing still) and the wind caused because of your own motion will be added.
When sailing downwind these two will be subtracted and you'd only feel the difference between them.
The wind you feel is called the apparent wind.
When driving your motorcycle I guess you either travel faster than the wind or not exactly in the same direction, which causes the apparent wind you still feel.
When you are moving in the same direction as the air with the same speed as the air, a circumstance also stated "with the same velocity as the air," the air is still in your frame of reference. Said another way: "in a coordinate frame oriented with you always at its origin, the air is not moving."
Therefore, you don't hear air moving around your ear. However, in the reference frame of a tree rustling in the wind or a sign that was blown over, the air certainly is moving. You would hear these phenomena unless your motorcycle is traveling faster than 1100ft/sec, which is the speed of sound in air, away from the sign, though not necessarily directly traveling directly away.
A fun digression is the question "whose reference frame is the absolute frame of reference?" "Relativity" is the answer to this. Relativity tells us the only absolute speed is the speed of light. All other observed motion and lack of motion is relative to some reference frame.
You experience this when you sail exactly downwind. The quietness is quite strange since you hear so much more wind noise when tacking, it is also the fastest the boat will travel. One big disadvantage is you enjoy no cooling from the wind and you can quickly warm up on a hot day.