The best way is to take a drive around town and see if there is anyone having a for rent sign on.....
Outside Buriram , direction Surin...a little after the 90 degree right turn,on your left is a little village called "Khe Ha" there is sometimes houses for rent at around 3000 a month,and only a 5 min drive to downtown Buriram....
Another good place is on the main street towards BigC , just after "W-mart" , there is a village called "Porn pong" there is also some people renting out houses and apartments....
In fact all the way in and around town there is plenty of nice little communities where u can rent a house or apartment
Claus