They said in advance of their trip that if necessary, they would take the search into their own hands.
All debris thought to be from the plane has so far been found in east Africa.
The plane disappeared on a flight from Kuala Lumpur to Beijing in March 2014 with 239 people on board.
It is thought that missing jet is most likely to have crashed into the southern Indian Ocean.
Campaigner Grace Subathirai Nathan, whose mother was on the plane, is making the self-funded journey with six other MH370 relatives - three from Malaysia, two from China and one from France.
"It has fallen into our hands to take on this search upon ourselves," she told reporters at Kuala Lumpur airport.
"After repeated requests for mobilisation of a search along the east African coastline, nothing has been done to date."
The families of those on board the flight say that the search for wreckage has not been systematic and that some possible findings appear to have been ignored.
The search for MH370 has concentrated off Australia's west coast, but no debris has been found there. It is due to be suspended early next year unless it makes a significant new discovery.
So far, at least six pieces have been found in east Africa that are considered certain or highly likely to have come from the missing Malaysia Airlines flight.
The relatives say they will hand out brochures when in Madagascar that will provide advice to villagers on how to recognise aircraft debris.
Their trip will last until 11 December, a statement posted on Facebook by the group Voice 370 said.
Voice 370 said last month that it had "no other choice except to take it upon ourselves to do something to find answers and closure".
Malaysia is in charge of co-ordinating the collection of any suspected debris, most of which has been sent to Australia for examination.
The Malaysian authorities in September responded to criticism of delays in retrieving suspected debris, arguing that they were acting on possible discoveries and that everyone should "allow the experts to conduct the verification processes".
Relatives of those who died on board Malaysia Airlines flight MH370 are travelling to Madagascar to lobby for the search for debris from the missing aircraft to be expanded.