The synthesis of RNA from a DNA template by RNA polymerase III (RNAP III), originating at a type 3 RNAP III promoter. A type 3 RNAP III promoter is composed of elements upstream of the transcription start site, including a TATA box. The human U6 snRNA gene has a type 3 promoter. Type 3 Pol III promoters have not been observed in S. cerevisiae.