Pretty much every car made in the last couple of decades has a gas hatch indicator arrow right on the dashboard:
Just look for the arrow next to the pump icon. I'd seen that little triangle thousands of times and never noticed it until a friend told me it indicates the side the gas cap is on.
If you're in a car that doesn't have a handy indicator, you can try driving up next to a structure with plate glass windows or another reflective surface.
If you angle it right, you should be able to see all of one side of your car reflected on the windows, and thereby check to see if you can find the gas door. If you can't see it, then it's probably on the other side, though you may want to do a U-Turn to confirm, especially if it's an unfamiliar car or you can't see the reflection very well.
This works best in shopping centers with parking lots, so you can get up pretty close to the glass and drive slowly or stop in front of it without blocking traffic.
Pull the fuel cap release lever under the drivers seat then look in your side mirrors.
When you release the outer fuel cap it springs outwards from the car body making it visible in one of the side mirrors.
EDIT: in response to Johnny's comment below - you should probably only do this while the vehicle is stationary, eg after you pull into the fuel stop. Apparently some people have been known to confuse the hood and fuel cover release.
Most cars will have the fuel cap on the drivers side of the vehicle. in America this is on the left side of the car. Cars made in America will all follow this standard, which means Chrysler, Ford and GM will generally all be on the left side of the vehicle.
In some other countries you drive on the opposite side of the road and therefore the driver is in the right side of the car, the fuel cap will then be on the right. When the cars are built for markets, such as American ones where we drive on the right side of the road, the cabin is retrofired such that the driver is placed on the left side of the vehicle but the gas cap remains on the right side of the car.
This of course isnt an exact science but I have found it to hold true for most cases.
TLDR; American made cars its on the left, foreign imported cars on the right.
Patterns: For single exhaust vehicles, the gas filling opening is on the opposite side of the exhaust, I've heard that this makes engineering the underbody easier: Japanese cars tend to have the exhaust on the right and American+German cars tend to have the exhaust on the left. Possible explanations for placing the filler opposite the driver include emergency fueling while pulled over. This also holds for placing the filler on the left for Japanese cars since Japan drives on the left 
.. then further down
Patterns for individual automakers (with a few exceptions, not all listed): It would be neat to see an infographic organizing this.
Japanese (90%+ left): Japanese cars are consistent within their model lines and even parent companies. The rate of exceptions is probably under 10% of vehicles produced in a given time period.
Honda/Acura: Left Toyota/Lexus: Left Nissan: Left (right: 350Z/G35 and 370Z/G37) Mazda: Left (Mazda3 right due to Ford influence) Mitsubishi: Left Subaru: Right
German: German automarkers tend to place the filler on the right, more consistently than American cars.
American: My knowledge of American cars is less, but they tend to be less consistent within their model lines and as a whole. Also, the relationship between parent companies is murkier.
Ford: Right (new Mustang is left, old Mustangs are right, Probe is left probably due to Mazda influence)
Chevrolet cars: Left.
Chevrolet Trucks/SUVs: Right
Dodge: Left (Viper is right)
Before the little arrow, here's what I was told:
If the gas gauge is on the right side of the instrument panel, then the gas cap is on the right. And vice versa - if the gas gauge is on the left side of the driver's instrument panel, then the gas cap is on the left.
It held true when I would drive the occasional rental car.
About the auto maker origin, American, Asian, European, you are partially right, the point is that Japanese drive on the left so they have the pump on the left and the Europeans drive on the right so they have the cap on the right. The exceptions came from joint ventures where the side of the cap depends on the origin of the car designer. For example: Asian designed cars like the toyota aygo that will also be also sold as the Citroën c2 in Europe. In Japan the same car branded toyota will have the wheel on the right and the cap on the left but if branded Citroën you will have the wheel on the left and the cap on the left side also as it is the same car, they just change the wheel side and sell it under different brands.
If there are no indicators, most people keep their car's manual in the car. The manual should tell you where the gas cap is, though it may be easier to just get out and check than to look through the book.