Parts of the Welsh bard's final resting place have been so badly damaged by the animals they have been left looking like a ploughed field.
St Martin's Church in Laugharne, West Wales, is visited by tourists from across the world thanks to the poet.
The simple white cross with Thomas's name written horizontally across the centre forms one part of a literary pilgrimage to the town.
Another is the boathouse, now a museum, where Thomas wrote Under Milk Wood and spent the last four years of his life.
Nearby is the celebrated Brown's Hotel, the watering hole where Thomas downed copious amounts of liquid inspiration.
Upset relatives whose loved ones are buried in the cemetery contacted their local MP when they discovered the damage.
Conservative Simon Hart said he was contacted when a family noticed extensive damage to their plot.
"I was contacted by a lady who has seen considerable damage done to her parents' grave," the Carmarthen West and South Pembrokeshire MP said.
"Understandably she is very upset about it and she is also worried that it will undermine the headstone.
"This is not that uncommon a problem in graveyards across the country but because badgers are so heavily protected it does cause a headache for church authorities.
"This is a very beautiful graveyard and obviously a place of great meaning, not only to Dylan Thomas fans who visit it from all over the world, but to families with loved ones buried there."
He added: "The badgers have caused a lot of damage, digging up the top few inches of soil so that it looks like a ploughed field in places."
Welsh Government animal experts from Aberystwyth have visited the site and confirmed that the damage has been caused by badgers.
One of the options open to the church is to apply for a licence to move the badgers.
"But this is expensive, a vet would have to be present and it's quite a logistical undertaking which may not even work," Mr Hart said.
As a result the church is looking at alternative ways of stopping the problem, such as improving fencing, to keep the badgers out.
A church spokesman said: "We are taking steps to prevent them from entering the graveyard."