Effective modal masses of one-dimensional modes are independent of the manner in which these modes are normalized. However, effective masses of multidimensional modes vary with the rotation of the coordinate system. In this paper, the concept of an effective mass tensor is introduced. It is supposed that effective masses should be recognized as tensors. It is shown that effective mass tensors indicate the inertial masses and principal axes of multidimensional modes, independent of the angle of the coordinate system. An example of a multidimensional structure response under a multidimensional earthquake shows the practical implications of an effective mass tensor.