Integral field spectroscopy (IFS) is a well-established method for measuring spectral intensity data of the form s(x,y,λ), where x, y are spatial coordinates and λ is the wavelength. In most flavors of IFS, there is a trade-off between sampling (x,y) and the measured wavelength band Δλ. Here we present the first, to our knowledge, attempt to overcome this trade-off by use of computational imaging and measurement diversity. We implement diversity by including a grating in our design, which allows rotation of the dispersed spectra between measurements. The raw intensity data captured from the rotated grating positions are then processed by an inverse algorithm that utilizes sparsity in the data. We present simulated results from spatial-spectral data in the experimental dataset. We used non-overlapping portions of the dataset to train our sparsity priors in the form of the dictionary, and to test the reconstruction quality. We found that, depending on the level of noise in the measurement, diversity up to a maximum number of measurements is beneficial in terms of reducing error, and yields diminishing returns if even more measurements are taken.