The understanding and predictive modeling of explosive blasts require advanced experimental diagnostics that can provide information on local state variables with high spatiotemporal resolution. Current datasets are predominantly based on idealized spherically symmetric explosive charges and point-probe measurements, although practical charges typically involve multidimensional spatial structures and complex shock-flow interactions. This work introduces megahertz-rate background-oriented schlieren tomography to resolve transient, three-dimensional density fields, as found in an explosive blast, without symmetry assumptions. A numerical evaluation is used to quantify the sources of error and optimize the reconstruction parameters for shock fields. Average errors are ∼3% in the synthetic environment, where the accuracy is limited by the deflection sensing algorithm. The approach was experimentally demonstrated on two different commercial blast charges (Mach ∼1.2 and ∼1.7) with both spherical and multi-shock structures. Overpressure measurements were conducted using shock-front tracking to provide a baseline for assessing the reconstructed densities. The experimental reconstructions of the primary blast fronts were within 9% of the expected peak values. The megahertz time resolution and quantitative reconstruction without symmetry assumptions were accomplished using a single high-speed camera and light source, enabling the visualization of multi-shock structures with a relatively simple arrangement. Future developments in illumination, imaging, and analysis to improve the accuracy in extreme environments are discussed.