Spectrometers provide our most detailed diagnostics of the solar coronal plasma, and spectral data is routinely used to measure the temperature, density, and flow velocity in coronal features. However, spectrographs suffer from a limited instantaneous field of view (IFOV). Conversely, imaging instruments can provide a relatively large IFOV, but extreme-ultraviolet (EUV) multilayer imaging offers very limited spectral resolution. In this paper, we suggest an instrument concept that combines the large IFOV of an imager with the diagnostic capability of a spectrograph, develop a new parametric model to describe the instrument, and evaluate a new method for "deconvolving" the data from such an instrument. To demonstrate the operating principle of this new slitless spectroscopy instrument, actual spectroscopic raster data from the Hinode/EUV Imaging Spectrometer (EIS) spectrometer is used. We assume that observations in multiple spectral orders are obtained, and then use a new inverse problem method to infer the spectral properties. Unlike previous methods, physical constraints and regularization derived from prior knowledge can be naturally incorporated as part of the solution process. We find that the fidelity of the solution is vastly improved compared to previous methods. The errors are typically only a few km s(-1) over a large IFOV, with a width of a few hundred pixels and an arbitrarily large height. These errors are not much larger than the errors in current slit spectroscopic instruments with limited IFOV. A further benefit is that the performance of candidate instruments can be optimized for specific scientific objectives. We demonstrate this by deriving optimum values for the spectral dispersion and signal-to-noise ratio.