Graphene solution-gated field-effect transistors (SGFETs) are a promising platform for the recording of cell action potentials due to the intrinsic high signal amplification of graphene transistors. In addition, graphene technology fulfills important key requirements for for in-vivo applications, such as biocompability, mechanical flexibility, as well as ease of high density integration. In this paper we demonstrate the fabrication of flexible arrays of graphene SGFETs on polyimide, a biocompatible polymeric substrate. We investigate the transistor's transconductance and intrinsic electronic noise which are key parameters for the device sensitivity, confirming that the obtained values are comparable to those of rigid graphene SGFETs. Furthermore, we show that the devices do not degrade during repeated bending and the transconductance, governed by the electronic properties of graphene, is unaffected by bending. After cell culture, we demonstrate the recording of cell action potentials from cardiomyocyte-like cells with a high signal-to-noise ratio that is higher or comparable to competing state of the art technologies. Our results highlight the great capabilities of flexible graphene SGFETs in bioelectronics, providing a solid foundation for in-vivo experiments and, eventually, for graphene-based neuroprosthetics.