You can’t actually fly to Koh Samui itself from Bangkok with budget airline Air Asia – but you can fly from Bangkok to Surat Thani and then get the ferry, which takes a bit longer but is somewhat cheaper if you don’t mind the extra hassle. Certainly it cuts out the tedious 12 hour overnight bus ride from Bangkok to Surat.
UPDATE: As of October 2012, Air Asia are now offering a Koh Samui Island Transfer service, where you buy a combined flight and ferry ticket. You fly into Surat Thani and then are transferred to the ferry which takes you to Samui, thus making it much more convenient and hassle free. See BangkokToSamui.com for full details.
For details about organising the transfer and ferry yourself, see this page for info on getting cheap flights to Surat and then the ferry to Koh Samui.
The only way to fly direct from Bangkok to Koh Samui is with Bangkok Airways or Thai Airways – check Skyscanner to compare the fares and flight times of the two airlines.
The flight to Surat Thani is also useful if you want to get to Koh Phangan or Koh Tao. I found this first hand write up of getting to Phangan this way at Greg’s Travel Tips.