If the phone was originally an AT&T phone, then no you can't do that.
AT&T is a GSM network. So only GSM phones will work on that network. That means your phone is GSM. While Straight Talk has some phones that work on GSM and others that are CDMA (depends on the area you are in and if they are using AT&Ts or Sprints network) they are not convertible or interchangeable.
You would need to get a phone that comes from a CDMA network to get it to work for you.