In this Letter, we propose spatially incoherent multiwavelength three-dimensional (3D) microscopy that exploits holographic multiplexing and is based on computational coherent superposition (CCS). The proposed microscopy generates spatially incoherent wavelength-multiplexed self-interference holograms with a multiband-pass filter and spatially and temporally incoherent light diffracted from specimens. Selective extractions of 3D spatial information at multiple wavelengths from the holograms are realized using the CCS scheme. We constructed fully mechanical-motion-free holographic multiwavelength 3D microscopy systems and conducted experiments to demonstrate the microscopy.