Intense and complex instrumental backgrounds, against which the much smaller signals from celestial sources have to be discerned, are a notorious problem for low- and intermediate-energy {gamma}-ray astronomy (~50keV-10MeV). Therefore, a detailed qualitative and quantitative understanding of instrumental line and continuum backgrounds is crucial for most stages of {gamma}-ray astronomy missions, ranging from the design and development of new instrumentation through performance prediction to data reduction. We have developed MGGPOD, a user-friendly suite of Monte Carlo codes built around the widely used GEANT (ver. 3.21) package, to simulate ab initio the physical processes relevant for the production of instrumental backgrounds. These include the build-up and delayed decay of radioactive isotopes as well as the prompt de-excitation of excited nuclei, both of which give rise to a plethora of instrumental {gamma}-ray background lines in addition to continuum backgrounds. The MGGPOD package and documentation are publicly available online (http://sigma-2.cesr.fr/spi/MGGPOD/). We demonstrate the capabilities of the MGGPOD suite by modeling high-resolution {gamma}-ray spectra recorded by the Transient Gamma-Ray Spectrometer (TGRS) on board Wind during 1995. The TGRS is a Ge spectrometer operating in the 40keV-8MeV range. Because of its fine energy resolution, these spectra reveal the complex instrumental background in formidable detail, particularly the many prompt and delayed {gamma}-ray lines. We evaluate the successes and failures of the MGGPOD package in reproducing TGRS data and provide identifications for the numerous instrumental lines.