You CAN bring a dog or cat into Thailand.  It is suggested that you use a professional pet transporter to facilitate the paperwork.  You will need:

• Authorization to import your pet by air  (Ror6 Form if you are in transit, and Ror7 Form if you are staying in Thailand) You get these at the International Airport in Bangkok.
• A special importation form (Kor Sor Kor 102)
• Identity certificate signed in your country of origin by a registered veterinarian, describing the age, breed, sex, color and any specific markings on your pet.
• A rabies vaccination certificate, showing the vaccination took place at least 15 days prior to your departure. 
• A leptospirosis vaccination certificate- showing the vaccination took place at least 21 days prior to departure.
• Your animal must be healthy and free from any sign of infectious or contagious disease.
• Your animal must have resided in your home country at least 6 months before your departure date.

There will be fees and import duties to pay.  If you are in transit, or are staying less than 6 months, your pet will be exempted from import duties.  But you will have to fill out a guarantee form that the pet will be leaving, and you will pay a deposit on the guarantee at the airport.