Fabrication methods for contoured polyhedral cavities for particle acceleration are disclosed. The process may include: trimming flat sheets to a conformal shape; bending the sheets to form a contour that is axially curved and azimuthally flat; and joining the sheets to form a circumferentially polyhedral cavity that is configured to support a resonant electromagnetic field at cryogenic temperatures. The resulting cavity may have ductile or even brittle superconducting materials with an axially-oriented grain structure at each point on the circumference of the cavity. As part of the assembly process, the sheets may be bonded to a supporting substrate of thermally conductive material having integrated cooling passages. The supporting substrates may be configured to have electrical contact near the cavity openings while having a small gap near the equators of the cavity. Moreover, mode-coupling channels and waveguides may be provided to extract energy from undesired deflection modes.