We report measurements of the In-115 nuclear spin-lattice relaxation rate (1/T-1) between T = 0.09 and 100 K in the new heavy fermion (HF) compound CeIrIn5. Ar 0.4 less than or equal to T less than or equal to 100 K, 1/T-1 is strongly T-dependent, which indicates that CeIrIn5 is much more itinerant than known Ce-based HFs. We find that I/T1T, subtracting that for LaIrIn5, follows a (1 / tau + theta)(3/4) variation with 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 hulk superconductivity sets in at 0.40 K below which the coherence peak is absent and 1/T-1 follows a T-3 variation, which suggests unconventional superconductivity with Line-node gap.