  We solve closed string theory in all regular homogeneous plane-wave
backgrounds with homogeneous NS three-form field strength and a dilaton. The
parameters of the model are constant symmetric and anti-symmetric matrices
k_{ij} and f_{ij} associated with the metric, and a constant anti-symmetric
matrix h_{ij} associated with the NS field strength. In the light-cone gauge
the rotation parameters f_{ij} have a natural interpretation as a constant
magnetic field. This is a generalisation of the standard Landau problem with
oscillator energies now being non-trivial functions of the parameters f_{ij}
and k_{ij}. We develop a general procedure for solving linear but non-diagonal
equations for string coordinates, and determine the corresponding oscillator
frequencies, the light-cone Hamiltonian and level matching condition. We
investigate the resulting string spectrum in detail in the four-dimensional
case and compare the results with previously studied examples. Throughout we
will find that the presence of the rotation parameter f_{ij} can lead to
certain unusual and unexpected features of the string spectrum like new
massless states at non-zero string levels, stabilisation of otherwise unstable
(tachyonic) modes, and discrete but not positive definite string oscillator
spectra.
