"Until we find out what happened, we cannot move on, we cannot grieve, and it consumes our thoughts 24/7; it's a harrowing, distressing and frustrating road.
"We as a group are solely committed to finding our loved ones and so far we have been dissatisfied with the investigation and with the release of the five-page preliminary report that only raised more questions than it answered, so we hope the authorities take heed of our requests."
The letter, signed by Mrs Weeks and relatives of 10 other victims, was sent on Thursday.
It said: "Our purpose is to support each other, with the sole intention of finding MH370 and our loved ones.
"Due to the lack of physical evidence that MH370 ended in the southern Indian Ocean, the families are in urgent need for the conclusion based on Inmarsat data analysis that the aircraft's flight ended in that ocean to be reconsidered to confirm its accuracy. Further, if Inmarsat's analysis is unable to rule out other flight paths as a possibility, that fact must be acknowledged.
"Given the lack of tangible evidence of what happened to MH370, in our view data analysis that only indicates a probably southern flight path is an insufficient basis to support a definitive conclusion that no other flight path was possible."
Inmarsat is a company whose satellite network was used to help deduce the supposed final route of MH370. That data was a major factor in the search being moved from Malaysia to an area four hours' flying time off Western Australia.
Voice 370 is "imploring" authorities to release the raw satellite data "so that it can be subject to broader analysis by relevant experts".
"The satellite data is the only lead we have and is key in identifying MH370's flight path ... In view of the lack of emergency locator transmitter activation, zero detected debris and the lack of convincing 'pings', we feel it is necessary that the data be subject to independent third-party review.
"It is our hope that with out-of-box thinking the whole world can help to look for the plane."
The group has also asked that any non-governmental search organisations contracted to hunt for MH370 be overseen by the Woods Hole Oceanographic Institution (WHOI), the world's largest private, non-profit oceanographic research group.
"In our view WHOI is not a commercial entity and its successful location of Air France 447 demonstrates that it has the experience and expertise to conduct the search for MH370 in an ethical manner."
The group thanks all the countries involved in the search for their "unrelenting efforts to help" so far.
Mr Weeks, 37, was travelling to Mongolia to start a new job as a mechanical engineer. He leaves sons Lincoln, 3, and Jackson, 1.
Auckland businessman Ximin Wang, 50, was also on the flight.
