Miniaturized, energy-efficient and application-specific spectral sensing systems promise to be a highly sought-after technology in the coming years, with potential applications in areas such as: distributed sensor systems, IoT devices, mobile autonomous platforms, and many others. We present in this work the design, construction and measurement results of a compact, mid-infrared spectrometer working in the 3 - 4 µm spectral region, attractive for applications requiring the identification of polymer materials. The spectrometer is based on linear-variable filters (LVF) combined with an uncooled HgCdTe linear-detector array (LDA). The design and architecture of the device is described and discussed in the context of miniaturization challenges and constraints. Measured spectra of thin polyimide and polystyrene foils are presented to prove the applicability of the developed device to polymer materials detection and identification.