Conductive, flexible graphene/poly(trimethylene carbonate) (PTMC) composites were prepared. Addition of just 3 wt % graphene to PTMC oligomers functionalized with methacrylate end-groups followed by UV cross-linking resulted in more than 100% improvement in tensile strength and enhanced electrical conductivity by orders of magnitude without altering the processability of the host material. The addition of graphene also enhanced mesenchymal stem cell (MSC) attachment and proliferation. When electrical stimulation via the composite material was applied, MSC viability was not compromised, and osteogenic markers were upregulated. Using additive fabrication techniques, the material was processed into multilayer 3D scaffolds which supported MSC attachment. These conducting composites with excellent processability and compatibility with MSCs are promising biomaterials to be used as versatile platforms for biomedical applications.