We present a field-portable lensfree tomographic microscope, which can achieve sectional imaging of a large volume ( 20 mm3) on a chip with an axial resolution of <7 ?m. In this compact tomographic imaging platform (weighing only 110 grams), 24 light-emitting diodes (LEDs) that are each butt-coupled to a fibre-optic waveguide are controlled through a cost-effective micro-processor to sequentially illuminate the sample from different angles to record lensfree holograms of the sample that is placed on the top of a digital sensor array. In order to generate pixel super-resolved (SR) lensfree holograms and hence digitally improve the achievable lateral resolution, multiple sub-pixel shifted holograms are recorded at each illumination angle by electromagnetically actuating the fibre-optic waveguides using compact coils and magnets. These SR projection holograms obtained over an angular range of ±50° are rapidly reconstructed to yield projection images of the sample, which can then be back-projected to compute tomograms of the objects on the sensor-chip. The performance of this compact and light-weight lensfree tomographicmicroscope is validated by imaging micro-beads of different dimensions as well as a Hymenolepis nana egg, which is an infectious parasitic flatworm. Achieving a decent three-dimensional spatial resolution, this field-portable on-chip optical tomographic microscope might provide a useful toolset for telemedicine and high-throughput imaging applications in resource-poor settings.