A grand challenge in designing polymeric materials is to tune their properties by macromolecular engineering. In this context, one of the drawbacks that often limits broader applications of polymers under high temperature conditions is their poor thermal conductivity kappa. Using molecular dynamics simulations, we establish a structure-property relationship in hydrogen-bonded polymer blends for possible improvement of kappa. For this purpose, we investigate two experimentally relevant hydrogen-bonded systems: one system consists of short poly(N-acryloyl piperidine) (PAP) blended with longer chains of poly(acrylic acid) (PAA), and the second system is a mixture of PAA and short polyacrylamide (PAM) chains. Simulation results show that PAA-PAP blends are at the onset of phase separation over the full range of the PAP monomer mole fraction phi(PAR) which intensifies even more for phi(PAP) > 0.3. While PAA and PAP interact with preferential hydrogen bonding, phase separation is triggered by the dominant van der Waals attraction between the hydrophobic side groups of PAP. However, if PAP is replaced with PAM, which has a similar chemical structure as PAP without the hydrophobic side group, PAA-PAM blends show much improved solubility. Better solubility is due to the preferential hydrogen bonding between PAA and PAM. As a result, PAM oligomers act as cross-linking bridges between PAA chains resulting in a three-dimensional highly cross-linked network. While kappa for PAA-PAP blends remain almost invariant with phi(PAR), PAA-PAM systems show improved kappa with increasing PAM concentration and also with respect to PAA-PAP blends. Consistent with the theoretical prediction for the thermal transport of amorphous polymers, we show that kappa is proportional to the materials' stiffness, that is, the bulk modulus K and sound velocity v of PAA-PAM blends. However, no functional dependence between kappa and K (or v) is observed for the immiscible PAA-PAP blends.