Actomyosin machinery is a fundamental engine consisting mostly of actin filaments, molecular motors, and passive cross-linkers, generating mechanical forces required for biological processes of non-muscle cells such as cell migration, cytokinesis, and morphogenesis. Although the molecular and physical properties of key elements in the actomyosin machinery have been characterized well, it still remains unclear how macroscopic force buildup and dissipation in actomyosin networks and bundles depend on the microscopic properties of individual cytoskeletal components and their local interactions. To bridge such a gap between macroscopic and microscopic scales, we have developed a three-dimensional computational model of actomyosin bundles clamped to an elastic substrate with minimal components: actin filaments, passive cross-linkers, and active motors. Our model accounts for several key features neglected by previous studies despite their significance for force generation, such as realistic structure and kinetics of the motors. Using the model, we systematically investigated how net tension in actomyosin bundles is governed via interplay between motors and cross-linkers. We demonstrated motors can generate large tension on a bundle in the absence of cross-linkers in a very inefficient, unstable manner. Cross-linkers help motors to generate their maximum potential forces as well as enhance overall connectivity, leading to much higher efficiency and stability. We showed further that the cross-linkers behave as a molecular clutch with tunable friction which has quite distinct effects on net tension depending on their cross-linking angles. We also examined the source of symmetry breaking between tensile and compressive forces during tension generation process and discussed how the length and dynamics of actin filaments and the stiffness of the elastic substrate can affect the generated tension.