In this study, a label-free multi-resonant graphene-based biosensor with periodic graphene nanoribbons is proposed for detection of composite vibrational fingerprints in the mid-infrared range. The multiple vibrational signals of biomolecules are simultaneously enhanced and detected by different resonances in the transmission spectrum. Each of the transmission dips can be independently tuned by altering the gating voltage applied on the corresponding graphene nanoribbon. Geometric parameters are investigated and optimized to obtain excellent sensing performance. Limit of detection is also evaluated in an approximation way. Besides, the biosensor can operate in a wide range of incident angles. Electric field intensity distributions are depicted to reveal the physical insight. Moreover, another biosensor based on periodic graphene nanodisks is further proposed, whose performance is insensitive to the polarization of incidence. Our research may have a potential for designing graphene-based biosensor used in many promising bioanalytical and pharmaceutical applications.