Holthuijsen,The free rides are not restricted to Yamanashi residents or even Japanese, but they are only offered at certain times of year and require advance application by postcard.
The vistor centre at the site is however open daily (except Mondays), and there is an observation gallery to watch the trains if they are being tested.
Testing is conducted most days, but the schedule is given in Japanese on the site below. No details of exact times, though.
http://linear.jr-central.co.jp/schedule/index.html