Call a travel agent and book your trip through them. If you only need plane tickets, they can still arrange that for you. They will tell you how much money is due when. The beauty of this is that you can go to their office, sometime before the due date, and make a payment in cash. The travel agent will give you a receipt.
Go directly to the airlines ticket counter. You will need to make a trip to the airlines ahead of time. If you go to book your ticket the day of your trip, the plane may be full. Go directly to the ticket counter, book the flight and then pay in cash.
Use Western Union (see "Resources" below). Many airlines are aligned with Western Union including American Airlines. You can book your tickets and then have Western Union send them the cash. You can do this at one of Western Unions offices, or you can do it online with your debit card which is the same as using cash.
Check the specific airlines website. Companies such as American Airlines will allow you to pay for your tickets online. The airline gets your permission to withdrawal the cash from your checking account. This way you pay in cash, but you don't have to make the trip down to the airline.
