A method for constructing explicit marching-on-in-time (MOT) schemes to solve the time domain magnetic field volume integral equation (TD-MFVIE) on inhomogeneous dielectric scatterers is proposed. The TD-MFVIE is cast in the form of an ordinary differential equation (ODE) and the unknown magnetic field is expanded using curl conforming spatial basis functions. Inserting this expansion into the TD-MFVIE and spatially testing the resulting equation yield an ODE system with a Gram matrix. This system is integrated in time for the unknown time-dependent expansion coefficients using a linear multistep method. The Gram matrix is sparse and well-conditioned for Galerkin testing and consists of only four diagonal blocks for point testing. The resulting explicit MOT schemes, which call for the solution of this matrix system at every time step, are more efficient than their implicit counterparts, which call for inversion of a fuller matrix system at lower frequencies. Numerical results compare the efficiency, accuracy, and stability of the explicit MOT schemes and their implicit counterparts for low-frequency excitations. The results show that the explicit MOT scheme with point testing is significantly faster than the other three solvers without sacrificing from accuracy.