Sinn Fein’s Letterkenny councillor Gerry McMonagle says thieves struck at both Leck Graveyard and Conwal Graveyard on Sunday afternoon.
At the Leck incident a mother and daughter returned to their car to find the window smashed.
Thieves had made off with their handbags which contained personal items belonging to their deceased loved ones.
A number of items were also stolen in the Conwal car break-in.
Said Cllr McMonagle: “This is a particularly despicable crime and shows the contempt that these heartless thieves have for the rule of law.
“Preying on people attending their loved ones’ graves must be the lowest of the low when it comes to criminal behaviour.
“These people must be caught and feel the full rigours of the law. I would ask anyone who seen anything suspicious at Leck Graveyard on Sunday afternoon around 4-30pm to contact the Gardaí.”