We report measurements of the 115In nuclear spin-lattice relaxation rate ( 1/T1) between T = 0.09 and 100 K in the new heavy fermion (HF) compound CeIrIn5. At 0.4 < or = T< or = 100 K, 1/T1 is strongly T-dependent, which indicates that CeIrIn5 is much more itinerant than known Ce-based HFs. We find that 1/T1T, subtracting that for LaIrIn5, follows a (1 / T+straight theta)3/4 variation with straight theta = 8 K. We argue that this novel feature points to anisotropic, due to a layered crystal structure, spin fluctuations near a magnetic ordering. The bulk superconductivity sets in at 0.40 K below which the coherence peak is absent and 1/T1 follows a T3 variation, which suggests unconventional superconductivity with line-node gap.