Robustness of biochemical systems has become one of the central questions in Systems Biology, although it is notoriously difficult to formally capture its multifaceted nature. Maintenance of normal system function depends not only on the stoichiometry of the underlying interrelated components, but also on a multitude of kinetic parameters. For given parameter values, recent findings have aimed at characterizing the property of the system components to exhibit same concentrations in the resulting steady states, termed absolute concentration robustness (ACR). However, the existing method for determining system components exhibiting ACR is applicable only to one class of mass-action networks for which this property can be confirmed, but not discarded. Here we design a new method which relies on biochemical network decompositions into subnetworks, called elementary flux modes, to identify ACR in a broader class of mass-action networks by using only the given stoichiometry. This approach reduces the problem of determining ACR to that of solving parameterized systems of linear equations, rendering it amenable to networks of larger sizes. Our unified framework will be helpful in analyzing this biologically important type of robustness as well as detection of novel systemic properties independent of the kinetic parameters for more complex biochemical networks.