Abstract We describe an algorithm for application of the classic “drizzle” technique to produce 3D spectral cubes using data obtained from the slicer-type integral field unit (IFU) spectrometers on board the James Webb Space Telescope. This algorithm relies upon the computation of overlapping volume elements (composed of two spatial dimensions and one spectral dimension) between the 2D detector pixels and the 3D data cube voxels, and is greatly simplified by treating the spatial and spectral overlaps separately at the cost of just 0.03% in spectrophotometric fidelity. We provide a matrix-based formalism for the computation of spectral radiance, variance, and covariance from arbitrarily dithered data and comment on the performance of this algorithm for the Mid-Infrared Instrument’s Medium Resolution IFU Spectrometer. We derive a series of simplified scaling relations to account for covariance between cube spaxels in spectra extracted from such cubes, finding multiplicative factors ranging from 1.5–3 depending on the wavelength range and kind of data cubes produced. Finally, we discuss how undersampling produces periodic amplitude modulations in the extracted spectra in addition to those naturally produced by fringing within the instrument; reducing such undersampling artifacts below 1% requires a four-point dithering strategy and spectral extraction radii of 1.5 times the point-spread function FWHM or greater.