Composite sheets based on aluminum alloys are of great interest for the aerospace and automotive industries. Due to the difficulty of joining dissimilar aluminum alloys by cold plastic deformation, in particular cold roll bonding (CRB), composites are more often obtained by hot roll bonding. In the present study, a novel technology to produce a promising AA5154/AA2024/AA5154 composite with strong bonding between layers was developed. Tensile shear tests, microstructural, and fractographic analyses allow one to evaluate the influence of the most important CRB parameters on the bond strength between aluminum alloys: pretreatment parameters of the surfaces to be bonded, CRB schedule, and post- and intermediate heat treatment. These factors are turned out to be crucial for the bond formation between materials. The developed technology for producing AA5154/AA2024/AA5154 composites provides a shear-bond strength value of 168 MPa, which is comparable to the shear strength value of the deformed AA2024 alloy. The present study also provides comprehensive data for the process of joining dissimilar aluminum alloys AA5154 and AA2024, which can be used for multiscale FE modeling.