We investigate three-dimensional resonant tunnelling (RT) through a do
uble potential barrier with a periodic modulation of the quantum well
in the plane of the layers. Both elastic and phonon-assisted RT throug
h transverse minibands separated by minigaps are considered. By using
the Breit-Wigner-like formula for the transmission probability we find
a more abrupt upward portion (front) of the current-voltage character
istics and a broader negative differential resistance region than in '
conventional' RT through double barrier heterostructures. Phonon-assis
ted tunnelling characteristics follow the same trends.