The simplest way is to buy a prepaid sim card which gives you a local Thailand number that can make and receive international calls. Most cell phone operators are GSM but Hutchison CAT does CDMA so whichever phone you have now can be used in Thailand. Your phone must be unlock (not restricted to one operator here in the USA), there are phone service guys in Thailand that will unlock your phone for a small fee.
The sim card will cost about $US 30~40, you can then buy recharge cards when you run out of the initial credits. You can buy the sim cards almost everywhere, just look for anyplace selling new/used mobile phones, they are everywhere. These places also know how to unlock your phone if you need that done.