Some parish priests struggling to keep up with graveyard maintenance say they are facing increasing hostility.
One priest, the Reverend Nia Williams of Bethesda, Gwynedd, told BBC Radio Wales' Eye On Wales programme she had been harassed after being unable to keep up with mowing the grass.
She said she had had nuisance phone calls at night, and a threat to "do her in".
A Church in Wales survey has found that 20% of its graveyards are in disrepair.
The programme found that Ms Williams' experiences were not unique. Other parish priests have reported feeling vilified because of the state of their graveyards.
Ms Williams is responsible for maintaining five graveyards across three parishes.
In the case of the Glanogwen churchyard, she said woodland had almost completely taken it over.
In the Llanllechid churchyard, saplings could be seen growing out of old box graves.
"Nobody wants to see graves in that state," she said. "But it's the cost of maintenance and getting people to volunteer to do it which is the problem.
"It's not a lack of wanting to do something on our part. It's just the financial burden."
The cost of mowing a graveyard is £700 to £1,000, depending on the size of the plot, and how difficult it is to get around the headstones.
Tree surgery and keeping boundary walls safe only add to the cost.
As well as trying to keep up with mowing, Gwynedd council has asked Ms Williams to repair a section of stone wall which has fallen down on to a public footpath.
It is unlikely that the council will help with the cost, she said.
The situation in Wales differs to England, where full - and therefore closed - Church of England graveyards are automatically transferred to local authority control.
In April, the Church in Wales flagged up the problem and approached the Welsh Assembly Government and Welsh Local Government Association (WLGA).
"We've had a few meetings but it's taking a bit of time and I accept that local government at the moment has a lot on its mind," said Alex Glanville, the author of the report.
"We want to talk more to the WLGA and to local authorities and community councils about the issue, because we think that it's only by talking and working together, we might find solutions.
"This isn't just about our problem".
Two-thirds of Church in Wales graveyards are expected to fill up over the next ten years, and the church is discouraging any further extensions because of their long-term liability.
This will add further pressure to municipal graveyards, some of which are running out of burial space.
Urban areas such as Cardiff face the most pressure.
Cardiff council is about to finish an extension at Thornhill cemetery and is in the process of planning for an extension at Western cemetery, which will add another 10 to 20 years of burial space.
According to Martin Birch, bereavement manager for Cardiff council, 25% of people are still choosing to be buried.
"The issue of grave space has been prominent for the last 10 years and it's rapidly running out, particularly in urban areas", he said.
"Time ticks by quickly. If I'm still here in15 years' time I will have to find more burial space, but the cost of land is premium."
Welsh Conservative justice and housing minister spokesman Mark Isherwood, who chairs the cross party group on funerals and bereavement, said everyone was entitled to appropriate and dignified resting place for their loved ones and for themselves.
"This is a ticking time bomb. If we don't address this now then we have the possibility or the likelihood of social injustice that will often impact on the weakest the hardest."
Ms Williams predicted that her last working graveyard would fill up over the next seven years.
"Sometimes I feel that I'm ministering more to the dying than I am to the living and that can be very frustrating," she said.
Eye On Wales is on BBC Radio Wales on Monday 31 May, at 1830 BST.