Rickey said he has even sat in the cemetery at night with a baseball bat hoping to catch them. He said he has talked with a couple of police officers but they are rotated in and out of the area very frequently.
Former caretaker Henry "Lou" Barnhart wrote a history of the three-acre cemetery, which has more than 2,000 graves, in 2002.
"Vandalism and the knocking over of gravestones has been a problem at the cemetery," he wrote. "In an attempt to curb this discouraging problem, a 7-foot high chain link fence was installed in 1974 along the rear and two sides of the cemetery. For a while there was some lessening of vandalism, but in time, fences were cut and the problem continues."
The cemetery is a significant one in Aberdeen, as is Grove Presbyterian Church, now on East Bel Air Avenue.
Reiber said it is the oldest cemetery in the city and is linked historically to the city's founders.
"The lineage of Aberdeen is well-represented there," he said.
The graveyard also contains at least 80 unidentified people buried there, Rickey said.
"There's a lot of unmarked graves," he said.
Grove Presbyterian Church dates itself to 1854, according to C. Milton Wright's "Our Harford Heritage."
Anyone with information on the crime may be eligible for a reward of up to $2,000 by contacting the Aberdeen police, 410-272-2121.