Many airlines fly to Bangkok, presenting travelers with many choices in getting there. We find that shopping around about three months in advance will usually produce the best option. From the United States we have flown on Korean Airlines, China Airlines, and Cathay Pacific (from Los Angeles or New York). These airlines had the best rates and good service, but its routes included stopovers in either Hong Kong, Seoul, or Taipei.
If you are in Europe, there are also many choices. Some airlines, like British Airways, fly direct, with daily service. For those based in Australia/New Zealand or Asia, flying over is no more than a hop, skip, and a jump. Best to check prices online before working with your travel agent for promotional fares.