Make a list of all the flights you've ever taken. If you're a frequent flyer, you may need to check through emails or credit card records to help you.
Visit the websites of the airlines you've travelled with. Most will have a map showing the destinations they fly to and the route their planes take to get there. You may need to click on a destination to view routes flown. There are also a number of online services available, such as Airlineroutemaps.com and Arm.64hosts.com, that supply route information from various airlines.
Work out which countries all of the flights you've ever taken have flown over and compile a list. If the maps provided online are not detailed enough, cross reference them with an atlas.