The statement by Voice 370, an international next-of-kin group, adds to doubts over the debris that have been expressed already by a number of individual family members.
"Needless to say, most families have refused to accept the Malaysian verdict, and are awaiting a more definite and conclusive analysis," the group said.
Malaysian Prime Minister Najib Razak announced last week that a wing part known as a flaperon, which washed up on the French island of Reunion, had been confirmed by experts in France as part of MH370.
Najib called it proof of his government's assertion that the jet met a disastrous end somewhere in the Indian Ocean, a position backed by many aviation experts.
But many MH370 family members remain deeply suspicious of Malaysia's handling of the aviation enigma and have rejected that conclusion, with some still harbouring the belief that the plane landed safely somewhere.
Voice 370 said Malaysia is the only party to have conclusively stated that the Reunion wreckage was from MH370.
French authorities have said there was a "very high probability" that the debris was linked to the Malaysia Airlines Boeing 777.
MH370 vanished on March 8, 2014 with 239 passengers and crew, triggering one of aviation's greatest mysteries. A search operation -- the biggest in history -- is still ongoing in the southern Indian Ocean.
Experts in France are examining the flaperon for any clues into what may have caused the aircraft to inexplicably veer off course en route from Kuala Lumpur to Beijing.
Authorities in Reunion, the island nation of Mauritius, and elsewhere in the Indian Ocean also have announced stepped-up vigilance for possible debris.
No further results of the flaperon analysis have been released, nor has any other debris been confirmed as from MH370.
Malaysia's government and the state flag carrier have faced intense criticism from next-of-kin who accuse both of a fumbled response to the jet's disappearance, confusing statements, and failing to share information with relatives, charges that are denied.
Voice 370 said families remain "apprehensive" about Malaysia's handling of the issue, and have "doubts about their expertise, capabilities and intentions".
It called for all potential MH370 debris to "be analysed at a reputable place with the appropriate expertise and equipment", including possibly the French government or air-safety authorities from other "advanced nations".
Australian authorities leading the search of the southern Indian Ocean seafloor for MH370 wreckage, along with other experts, believe the flaperon likely came from MH370.
The search operation's headquarters said in a statement Wednesday that it was "in all probability" from MH370.
Malaysia has said certain characteristics of the wing fragment, including its paint, matched MH370 maintenance records.
Scientists have also said barnacles on the flaperon could indicate how long it was in the water, and perhaps where it had been, information that could help to narrow the current massive search zone.
Search crews hope to locate a crash site and recover the plane's data recorders, which would be analysed for clues on the cause of the disaster.