A simple, compact, and inexpensive tunable volume holographic imaging spectrometer is presented to acquire narrowband multispectral fluorescence measurements. It is constructed with three simple achromatic lenses, a transmission volume hologram, and a detector array. Experiments are carried out to evaluate the performance of this spectrometer. The experimental results indicate that the tunable spectrometer can operate at a wavelength range from 400 to 700 nm. Additionally, this spectrometer can provide a lateral field-of-view greater than 33 mm and a spatial resolution of 1.0 mm under narrow band illumination (full-width-half-maximum bandwidth 20 nm). Multispectral fluorescence molecular tomography experiments are conducted with this spectrometer to test the applicability of narrowband multispectral macroscopy.