We present a simple method for continuous snapshot multispectral imaging or multispectral videography that achieves high-speed spectral video recording without the need for mechanical scanning and much computation for datacube construction. The enabling component of this method is an array of periscopes placed in a prism-based spectral shaper that spectrally separates the image without image deformation. As a proof-of-principle demonstration, we show five-color multispectral video recording in the visible range (200×200 pixels per spectral image frame) at a record high frame rate of at least 2800 frames per second. Our experimental results indicate that this method holds promise for various industrial and biomedical applications such as remote sensing, food inspection, and endoscopy.