Source analysis of magnetoencephalography (MEG) data requires the computation of the magnetic fields induced by current sources in the brain. This so-called MEG forward problem includes an accurate estimation of the volume conduction effects in the human head. Here, we introduce the Cut finite element method (CutFEM) for the MEG forward problem. CutFEM's meshing process imposes fewer restrictions on tissue anatomy than tetrahedral meshes while being able to mesh curved geometries contrary to hexahedral meshing. To evaluate the new approach, we compare CutFEM with a boundary element method (BEM) that distinguishes three tissue compartments and a 6-compartment hexahedral FEM in an n = 19 group study of somatosensory evoked fields (SEF). The neural generators of the 20 ms post-stimulus SEF components (M20) are reconstructed using both an unregularized and a regularized inversion approach. Changing the forward model resulted in reconstruction differences of about 1 centimeter in location and considerable differences in orientation. The tested 6-compartment FEM approaches significantly increase the goodness of fit to the measured data compared with the 3-compartment BEM. They also demonstrate higher quasi-radial contributions for sources below the gyral crowns. Furthermore, CutFEM improves source separability compared with both other approaches. We conclude that head models with 6 compartments rather than 3 and the new CutFEM approach are valuable additions to MEG source reconstruction, in particular for sources that are predominantly radial.