The understanding and predictive modeling of explosive blasts require advanced experimental diagnostics that can provide information on local state variables with high spatiotemporal resolution. Practical charges typically involve multidimensional spatial structures and complex flow-shock interactions (e.g., multiple shock interactions that constructively interfere). Here, MHz-rate background-oriented schlieren tomography (BOST) is developed to resolve transient, three-dimensional density fields, such as 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 MHz 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.