Your best bet to find the really cheap flights Bangkok to Los Angeles CA. is to start looking on Expedia and Travelocity which will give you a ballpark idea of the cheaper airfares once you have the cheapest flights that will work for your travel window, compare them you will see that they are usually pretty close. Then what you want to do is go to Side Step or Fare Compare to find the really cheap flights.

Recently I found a last minute discount flight on Side Step that saved me over $250 if i was willing to stay in LA for 2 days more than my ideal plan. Fare Compare also will come up with flights that you would not expect to see. On this particular trip I ended up flying with Asiana AIr and had a lay over in Inchon Korea just outside Seoul, but who cares which airports they run you through, as long as you get to where you are going, and get the really cheap airfares.

To find the the cheaper flights USA to Thailand, your best fares will usually be right around 60 days out, then the airlines start raising the prices, so a $600 dollar seat will escalate up to $1200 and higher. Today you will not find last minutediscount fares very often, and the days of half empty jets are coming to a close, as the software that the airlines use can anticipate the demand fairly well nowadays.
The really cheap flights LA to Bangkok or Bangkok to LA run around $650 to $800 on both Eva air and China Air. Some people think that the service on Eva is better I find no difference myself, however the big negative for China Air is that you need to lay over in Taipei arguably the most boring airport in Asia.