Many nanometre-sized building blocks will readily assemble into macroscopic structures. If the process is accompanied by effective control over the interactions between the blocks and all entropic effects, then the resultant structures will be ordered with a precision hard to achieve with other fabrication methods. But it remains challenging to use self-assembly to design systems comprised of different types of building blocks-to realize novel magnetic, plasmonic and photonic metamaterials, for example. A conceptually simple idea for overcoming this problem is the use of 'encodable' interactions between building blocks; this can in principle be straightforwardly implemented using biomolecules. Strategies that use DNA programmability to control the placement of nanoparticles in one and two dimensions have indeed been demonstrated. However, our theoretical understanding of how to extend this approach to three dimensions is limited, and most experiments have yielded amorphous aggregates and only occasionally crystallites of close-packed micrometre-sized particles. Here, we report the formation of three-dimensional crystalline assemblies of gold nanoparticles mediated by interactions between complementary DNA molecules attached to the nanoparticles' surface. We find that the nanoparticle crystals form reversibly during heating and cooling cycles. Moreover, the body-centred-cubic lattice structure is temperature-tuneable and structurally open, with particles occupying only approximately 4% of the unit cell volume. We expect that our DNA-mediated crystallization approach, and the insight into DNA design requirements it has provided, will facilitate both the creation of new classes of ordered multicomponent metamaterials and the exploration of the phase behaviour of hybrid systems with addressable interactions.