We discuss bacterial biofilms and the scope for describing their viscoelastic mechanical properties as a consequence of their underlying polymeric and multiphase morphology. Biofilms are the most prevalent phenotype of bacteria in nature. Biofilms form under conditions common in industry and in the body. They are structurally heterogeneous on multiple scales. We argue that the resolution of microscale mechanical properties is essential to fundamental understanding of the fate of biofilms in situations of flow, including the human circulatory system. Because of this need for microscale characterization, we developed the flexible microfluidic rheometer to characterize the elastic modulus and relaxation time of bacterial biofilms. The biofilms studied here are bacterial communities of Staphylococcus epidermidis and Klebsiella pneumoniae. The microfluidic device exploits the response of a flexible, deforming membrane to characterize the viscoelasticity of the test material. Attributes of the device are its simple fabrication and operation as well as its ability to accept biofilms grown at biologically relevant shear rates in the microfluidic environment. We find that the static and temporal responses of the valve membrane, as quantified by confocal microscopy, agree well with the viscoelastic properties of a model gellan gum as modeled by finite element simulation. Measurement of steady-state deformation yields both the linear and non-linear elastic response of the biofilms. We also report the transient response of the PDMS membrane coupled to the biofilm when the system is subjected to a stress relaxation experiment. We track the membrane deformation with the aim of extracting the viscoelastic relaxation time of the soft biological solid.