How to clip raster in GRASS GIS?

For cropping a raster map according to a vector map, you can use r.mask which also supports vector maps.

Update 2018: Meanwhile there are v.clip and a new Addon available: r.clip

I know it's late, but for others who have the same question. Please try this:

# limit your computational region to polygon_vector
r.mask vect=polygon_vector

# clip big_raster to polygon_vector (MASK set by r.mask above)
r.mapcalc ex=clipped_raster=big_raster

# export
r.out.gdal input=big_raster output=clipped_raster.tif