We report Arecibo 21 cm absorption-emission observations to characterize the physical properties of neutral hydrogen (H I) in the proximity of five giant molecular clouds (GMCs): Taurus, California, Rosette, Mon OB1, and NGC 2264. Strong H i absorption was detected toward all 79 background-continuum sources in the ~60 × 20 square degree region. Gaussian decompositions were performed to estimate temperatures, optical depths, and column densities of the cold and warm neutral medium (CNM and WNM). The properties of individual CNM components are similar to those previously observed along random Galactic sightlines and in the vicinity of molecular clouds, suggesting a universality of cold H i properties. The CNM spin temperature (T s) histogram peaks at ~50 K. The turbulent Mach numbers of CNM components vary widely, with a typical value of ~4, indicating that their motions are supersonic. About 60% of the total H i gas is WNM, and nearly 40% of the WNM lies in thermally unstable regime 500–5000 K. The observed CNM fraction is higher around GMCs than in diffuse regions, and increases with increasing column density (NH I) to a maximum of ~75%. On average, the optically thin approximation (N*H I) underestimates the total column density by ~21%, but we find large regional differences in the relationship between NH Iand the required correction factor, f =NH I/N*H I . We examine two different methods (linear fit of f versus log10(N*H I) and uniform Ts) to correct for opacity effects using emission data from the GALFA-H I survey. We prefer the uniform Ts method because the linear relationship does not produce convincing fits for all subregions.