In this paper, we introduce the notions of I-3-convergent, I-3(star)-convergent and I-3-Cauchy triple sequences in a fuzzy normed linear space. Besides, we establish some basic results related to those notions. Furthermore, we define the concepts of I-3(star)-Cauchy sequence in fuzzy normed space and some relations between I-3-Cauchy sequence in a fuzzy normed space are shown.