Forensic tests are being carried out on the remains following the raid near the town of Nayagarh on Sunday.
It is thought the find could be linked to the killing of female foetuses - a practice fairly widespread in India.
According to one estimate, up to 10m female foetuses may have been aborted in the country in the past two decades.
India, where boys have traditionally been favoured over girls, banned gender selection and selective abortion in 1994.
A senior police office in Orissa said they raided an area outside Nayagarh following a tip-off.
The bags, full of skulls and bones, were buried inside a pit in a plot of land belonging to a maternity clinic owner.
The clinic owner and manager have now been taken into custody by the police.
Officials say it is not possible at this stage to say exactly how many bodies were found in the bags.
"We cannot put any numbers. We have to wait for the details of the investigation," BK Sharma, Orissa crime branch inspector general said.
Media reports last week said seven female foetuses had been thrown into a garbage dump in another area in the state.
The recovery prompted the state chief minister to order a wider inquiry.
Skewed sex ratio
Sunday's raid came as part of the drive begun by the special crime police across the state.
Officials say a detailed investigation is now under way and initial indications point to sex selective abortions.
Similar incidents have been reported from other parts of the country in recent months, raising concerns about India's dwindling female population.
Female foeticide is believed to be one of the main reasons for the skewed sex ratio in India.
India's minister for Women and Child Development, Renuka Chowdhary, has said the government plans to monitor all pregnancies to prevent female foeticide.
In June, police arrested the owner of a clinic near Delhi on suspicion of illegally aborting female foetuses.
The arrest followed a raid at the clinic in Gurgaon in Haryana state. Officials said the remains of several foetuses were found in a well nearby.
Last year, a surprise raid in Patiala district in Punjab province unearthed the remains of at least 50 female foetuses from a well by a private clinic."
What's amazing is not the fact that this occurs, but that it has just now begun to make news. These. Are. Traditionally. Horrid. Awful. "People".