The authors study the multigrid waveform relaxation method (MGWRM) for solving initial value problems of the form \[ {du \over dt} + L_h u = f, \quad t > 0, \quad u(0) = u_0, \text{ with }L_h \in\mathbb{C}^{n \times n}, \tag{1} \] typically resulting from the spatial discretization of parabolic initial-boundary value problems on some grid with the discretization parameter \(h\). In the MGWRM, the waveform relaxation method \[ {du^{(\nu)} \over dt} + M_h u^{(\nu)} = N_h u^{(\nu - 1)} + f,\quad t > 0,\;u^{(\nu)}(0) = u_0,\;\nu = 1, \dots, \nu_1,\tag{2} \] serves as smoother, where \(L_h = M - N\). The coarse grid problem is also an initial value problem of the form (1) with the defect \(d_H := I^H_h (du^{(\nu_1)} /dt + L_h u^{(\nu_1)} - f)\) as right-hand side. The authors give a convergence analysis of the MGWRM with respect to the convergence of the corresponding multigrid method for the steady-state problem \(L_h u = f\). In order to get realistic convergence rate bounds the authors use the Fourier-Laplace analysis in analogy to the Fourier analysis in the steady-state case.