  We give a systematic treatment of lattice Green functions (LGF) on the
$d$-dimensional diamond, simple cubic, body-centred cubic and face-centred
cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three
lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there
is a close connection between the LGF of the $d$-dimensional hypercubic lattice
and that of the $(d-1)$-dimensional diamond lattice. We give constant-term
formulations of LGFs for all lattices and dimensions. Through a still
under-developed connection with Mahler measures, we point out an unexpected
connection between the coefficients of the s.c., b.c.c. and diamond LGFs and
some Ramanujan-type formulae for $1/\pi.$
