We present an optical tomographic diffractive microscope, a device able to image a complex refractive index distribution in three dimensions. Theoretical foundations are first recalled: diffraction under the first Born approximation explains the link between diffracted beam, object frequencies and physical properties of the object. We then describe our experimental setup, recording 2D interferograms in the image space, and detail the image reconstruction process underlying our tomographic microscope, which involves 2D transforms of the recorded interferograms, a peculiar 3D mapping of the data, and a final 3D Fourier reconstruction. We apply tomographic reconstruction to diatom skeletons, unicellular algae with cell walls made of silica, and compare it to holographic reconstruction. We further apply it to pollen grains and show differences between the real and imaginary parts of the measured complex refractive index. Finally, we also recall alternative tomographic methods.