Unsaturated polydienes are frequently hydrogenated to yield polyolefins that are more chemically stable. Here, the effects of partial hydrogenation on the phase behavior and nanostructure of polyisoprene-containing block copolymers are investigated. To ensure access to the order-disorder transition temperature (TODT) over a wide temperature range, we examine copolymers with at least one random block. Dynamic rheological and scattering measurements indicate that TODT increases linearly with increasing hydrogenation. Small-angle scattering reveals that the temperature-dependence of the Flory-Huggins parameter changes and the microdomain period increases, while the interfacial thickness decreases. The influence of hydrogenation becomes less pronounced in more constrained multiblock copolymers.