There is a bug (https://bugs.launchpad.net/ubuntu/+source/tzdata/+bug/1554806, not fixed at the time of writing this answer) in 16.04 which causes the contents of /etc/timezone to be overwritten with the old value when running dpkg-reconfigure -f noninteractive tzdata. The fix is as follows (from the above bug report):
$ sudo ln -fs /usr/share/zoneinfo/America/New_York /etc/localtime
$ sudo dpkg-reconfigure --frontend noninteractive tzdata
Current default time zone: 'America/New_York'
Local time is now: Mon Feb 20 07:30:33 EST 2017.
Universal Time is now: Mon Feb 20 12:30:33 UTC 2017.
$ cat /etc/timezone
America/New_York
No need to manually change the contents of /etc/timezone. This worked for me on Ubuntu 16.04.2 LTS.
