Natural self-assembled three-dimensional photonic crystals such as blue-phase liquid crystals typically assume cubic lattice structures. Nonetheless, blue-phase liquid crystals with distinct crystal symmetries and thus band structures will be advantageous for optical applications. Here we use repetitive electrical pulses to reconfigure blue-phase liquid crystals into stable orthorhombic and tetragonal lattices. This approach, termed repetitively applied field, allows the system to relax between each pulse, gradually transforming the initial cubic lattice into various intermediate metastable states until a stable non-cubic crystal is achieved. We show that this technique is suitable for engineering non-cubic lattices with tailored photonic bandgaps, associated dispersion and band structure across the entire visible spectrum in blue-phase liquid crystals with distinct composition and initial crystal orientation. These field-free blue-phase liquid crystals exhibit large electro-optic responses and can be polymer-stabilized to have a wide operating temperature range and submillisecond response speed, which are promising properties for information display, electro-optics, nonlinear optics, microlasers and biosensing applications.