I know I was very skeptical about several of the articles in February, which all seemed to originate with the Ministry of Culture of the Assad regime, and all squarely place the blame on the rebels ...
I'm not quite sure what's going on in most of Syria these days. The contents of Damascus Museum are safely in storage, and do not seem to have been looted. Palmyra was not looted last year, but has been shelled this year - the big temple now has holes in in, and sections of column drums in various colonnades have been damaged.
One reason it is hard to know what's going on is because the reputable journalists are having a hard time getting visas, when they do get into Syria they tend to be largely restricted to reporting from Damascus (although the conflict is moving increasingly into the capital), and, quite rightly, they are more interested in covering the huge human cost of the civil war.
This article seems to concentrate on Ebla:
Syrian War Devastates Ancient Sites - NYTimes.com
I've seen some of the satellite photos various groups on Facebook are using as "proof" of looting and ... well, they clearly can't read things like shadows and ... in short, although there are many wild claims going around about looting, mostly it seems to be in peoples' imaginations - those in Syria seem more interested in working out how to survive the war than how to steal bits of archaeology.
But yes, many historical monuments and archaeological sites have suffered badly from the cross-fire, and many have been all but destroyed.