"Looting" even of battlefield souvenirs is against the law (UCMJ).
Soldiers are briefed on this and the military is searching all
returnees.  When items are found, they are being confiscated and the
soldiers are apparently being prosecuted.  As far as I have seen, no
soldiers have been found with anything of "cultural" value, e.g. from
the various museums.  All the items that have been found so far have
been things like ashtrays, knives, and weapons either taken from Iraqi
troops or from the various palaces.  Soldiers are not saints, but so far
there is no evidence that they trashed any museums or archaeological
sites.