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".

The National Institute for Health and Care Excellence (NICE) said there were "major issues" in the way allergies were recorded and shared.
In its first guidelines on allergy, it is calling for known allergies to be recorded on all prescriptions.
The charity Allergy UK said a "lack of awareness" was putting lives at risk.
Official data recorded 18,079 incidents of drug allergies affecting patient safety in the NHS in England and Wales between 2005 and 2013.
Most involved medicines which were prescribed to a patient with a known allergy to that class of drugs.
Prof Mark Baker, the director of the centre for clinical practice at NICE, said: "If we know that giving someone a particular drug could cause them harm, or in the worst instances may even kill them, the utmost care must be taken to ensure they are not prescribed or administered that drug."
NICE said efforts were needed to "dispel confusion" over allergies after "major issues" were discovered as the guidelines were drawn up.
It called for improvements in the way doctors collect information on patients' allergies. It also wants a re-design of all prescriptions to include information on the classes of drugs to which a patient is known to be allergic.
Signs of allergy include fever, hives, wheezing, redness and swelling.
Dr Shuaib Nasser, a consultant allergist at Addenbrooke's Hospital in Cambridge, stated: "The guideline stresses the care all healthcare professionals must take when documenting new drug allergies and the importance of sharing this information with patients and other healthcare professionals.
"People should be provided with structured written information on drugs to avoid and be advised to check with their pharmacist before taking over-the-counter medicines.
"In some cases, specialist investigations will be required to confirm or exclude drug allergy."
Maureen Jenkins, clinical director at the charity Allergy UK, said: "Drug allergies can cause severe, sometimes life-threatening reactions but are often undiagnosed.
"We frequently have calls about the lack of awareness about drug allergy, inadequate documentation and communication between health professionals, which can put patients' lives at risk. We welcome these NICE guidelines, that will lay the foundation stone for better understanding and management of drug allergy across primary, secondary and tertiary care."

Boulden, 23, struck six birdies to card a six-under-par 68 after Australian Sarah Jane Smith took the lead with 67.
In a close contest, with the top 19 separated by just four shots, England's Jodi Ewar-Shadoff hit 71 and is four strokes back.
Scotland's Gemma Dryburgh and Michele Thomson are one shot behind Ewar-Shadoff after rounds of 69 and 74.
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.
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Thousands of lives are put at risk by doctors prescribing drugs to which patients are known to be allergic, the medicines watchdog says.
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Wales' Amy Boulden is three shots off the lead after the second round of the Women's Australian Open in Adelaide.