On a Mac, where QGIS uses the Kyngchaos.com GDAL 1.9 Complete framework, or on any platform where QGIS has been compiled against a GDAL build with support for the Google LIBKML driver, you can open .kmz files directly using the open dialog and selecting the
All Files (*) option, or simply drag/drop onto QGIS. (Otherwise, you will need to decompress the .kmz to .kml files as noted by @Kurt)
If the .kmz has multiple layers, you will be presented with a layer selection dialog (single layer .kmz files are immediately loaded):
You can select multiple layers in that dialog to import. The Example Overlay of Multiple Datasets .kmz file I tested (from here) allowed multiple layer import of point, line and polygon layer types.
EDIT: You can check for ogr LIBKML driver support in your version of QGIS by running the following commands in QGIS's Python Console:
print 'Has LIBKML ogr driver: ' + str(osgeo.ogr.GetDriverByName('LIBKML') is not None)