A new method for fast magnetic resonance imaging is presented. It provides a more rapid data acquisition than two-dimensional Fourier imaging (2DFI) by a factor which may be chosen depending on the required signal-to-noise ratio of the image. In addition to the readout gradient of 2DFI, the present method employs an oscillating modulation gradient. In this way, a curved alternating trajectory in k space is sampled after each spin excitation. For a p-times accelerated data acquisition, the trajectory consists of p periods, where p is of the order of 2 to 8 for low-frequency gradient modulation but can be chosen higher if certain hardware requirements are met. Adequate sampling density in k space is obtained by scanning shifted trajectories after subsequent spin excitations. The method can be combined with volume imaging (3DFI) and multiple slice 2DFI. It was implemented on a standard Philips Gyroscan system without any hardware modifications. Results obtained for an acceleration factor p = 4 are shown.