NII (or NIfTI) files are probably the most commonly-used format for multi-dimensional neuroimaging data as of this writing (late 2012). NIfTI is a raster format, with files generally containing at least 3-dimensional data: voxels, or pixels with a width, height, and depth. NIfTI files can be up to 7-dimensional: the first four dimensions are defined as 3 spatial dimensions and time, with the other dimensions used for "other stuff" -- though the time dimension is often to encode something other than time.
Most packages also support reading gzip-compressed NIfTI files, which should have the extension .NII.gz.
The format is extensively supported and documented by the NIH.