When flying around a thunderstorm, it is advised that you fly in the upwind portion of the thunderstorm. First off, how do you figure out which side is the upwind and downwind of a thunderstorm? Second, why fly in the upwind?
The "anvil" of a Thunderstorm is always on the downwind side. Airline pilots routinely deviate around the upwind side in hopes of a smoother ride.
You don't figure out which side of the storm (mature stage) has updrafts or downdrafts. You simply don't fly near, in, or under a thunderstorm. It's too dangerous. The FAA advises that you stay at least 20NM away from intense radar echos associated with thunderstorms in order to avoid clear air turbulence and hailstones. These factors may be present on all sides of a mature thunderstorm.
One reason that you stay away from the downwind side is hinted at by the funny quote by @psmears:
If you are downwind of a cheese factory, you will smell it, but if you are upwind of it, you won't.
Here is my take on it:
If you are downwind of a thunderstorm, you will get hit with hail, but if you are upwind of it, you won't.
Of course, it is possible there is no hail on the downwind side, but still it is a risk that is not usually necessary to take, and it is almost always possible to deviate as needed.
Below is an image taken from Professor Frank Ludlam's (Imperial College) 1961 article in Weather. The storm is moving right to left and affects are out to 20 km. You can see the large area of hail downwind.
I remember coming in to NAS Beeville in an A4 finishing up an instrument training hop. We were in a slow descent at 25k feet heading east at 250 kts. It was around sunset, with the sun behind and encroaching darkness in front of us. There were numerous large thunderstorm cells around us, some as high as 60-70k feet. As I looked up towards their tops I could see the lightning arcing from one side to the other of a cell in that weird, mesmerizing twilight. It was like a fireworks show, and quite impressive. There wasn't anybody out there that evening and we asked to deviate around weather. We got approval and did these lazy turns in our descent to the approach. One of those most pleasant fligths I visit often in my memory.
Simply put, flying downwind of thunderstorms increases your risk of hail.
The downwind portion of the thunderstorm is characterized by an anvil cloud. Typically the downwind side has a much higher probability of hail.
It is not uncommon for anvil clouds to extend a hundred miles down wind from larger cells. Hail may be can be obscured by precipitation. Hail can damage an aircraft even though other flight conditions such as turbulence are benign.