The hydride proton magnetic shielding tensors for a series of iridium(III) and rhodium(III) complexes are determined. Although it has long been known that hydridic protons for transition-metal hydrides are often extremely shielded, this is the first experimental determination of the shielding tensors for such complexes. Isolating the (1)H NMR signal for a hydride proton requires careful experimental strategies because the spectra are generally dominated by ligand (1)H signals. We show that this can be accomplished for complexes containing as many as 66 ligand protons by substituting the latter with deuterium and by using hyperbolic secant pulses to selectively irradiate the hydride proton signal. We also demonstrate that the quality of the results is improved by performing experiments at the highest practical magnetic field (21.14 T for the work presented here). The hydride protons for iridium hydride complexes HIrX2(PR3)2 (X = Cl, Br, or I; R = isopropyl, cyclohexyl) are highly shielded with isotropic chemical shifts of approximately -50 ppm and are also highly anisotropic, with spans (=δ11 - δ33) ranging from 85.1 to 110.7 ppm. The hydridic protons for related rhodium complexes HRhCl2(PR3)2 also have unusual magnetic shielding properties with chemical shifts and spans of approximately -32 and 85 ppm, respectively. Relativistic density functional theory computations were performed to determine the orientation of the principal components of the hydride proton shielding tensors and to provide insights into the origin of these highly anisotropic shielding tensors. The results of our computations agree well with experiment, and our conclusions concerning the importance of relativistic effects support those recently reported by Kaupp and co-workers.