Ford put one of three engines in your 1997 F-350 -- all of them V-8s. Two of the three are gasoline engines: a 5.8-liter or a 7.5-liter. The third is a 7.3-liter Powerstroke diesel. Each of the three engines has its own firing order.
The cylinder banks are the same even though the firing order is different between the two gasoline engines. From front to back, the passenger-side bank is numbered 1, 2, 3, 4 and the driver-side bank is numbered 5, 6, 7, 8. The firing order on the 5.8-liter engine is 1-3-7-2-6-5-4-8. The firing order on the 7.5-liter engine is 1-5-4-2-6-3-7-8. If you're not sure which engine is in your F-350, look at the vehicle identification number. The eighth position in the VIN identifies the engine. If the character is an "H," you have the 5.8-liter engine. If it's an "F," you have the 7.5-liter.
You know what kind of fuel you put in your F-350, so if you have the diesel engine, there is no need to decode the VIN. From front to back, the passenger-side bank is numbered 1, 3, 5, 7 and the driver-side bank is numbered 2, 4, 6, 8. The firing order on your 7.3-liter Powerstroke is 1-2-7-3-4-5-6-8.