A three-dimensional photonic crystal with a "woodpile" structure has been constructed. The sample is characterized experimentally as well as computationally in terms of reflection and transmission properties. A full band-gap appears at 10.9-13.1 GHz, which is the first gap, resulting in strong reduction of transmitted radiation. The crystal has properties characteristic of an effective dielectric medium at frequencies below the first band-gap. The frequency range above the first band-gap contains a series of bands interspersed with narrow gaps. The reflection and transmission results are complemented by calculations of the fields and power density within the crystal using the finite-difference time-domain (FDTD-) method. The power density is strongly reduced at frequencies which coincide with the first band-gap, in agreement with the reflection and transmission measurements. Close to the band-gap edges the field within the crystal have properties characteristic of standing wave modes in photonic band-gap materials.