The Josephson vortex (JV) lattice is a periodic array that scatters electromagnetic waves in the THz-frequency range. We show that JV lattices can produce a photonic band-gap structure (THz photonic crystal) with easily tunable forbidden zones controlled by the in-plane magnetic field. The scattering of electromagnetic waves by JVs results in a strong magnetic-field dependence of the reflection and transparency. Fully transparent or fully reflected frequency windows can be conveniently tuned by the in-plane magnetic field. These proposals are potentially useful for controllable THz filters.