  We find a general class of pp-wave string solutions with NS-NS $H_3$ or R-R
$F_3$ field strengths, which are analogous to solutions with non-constant $F_5$
recently considered by Maldacena and Maoz (hep-th/0207284). We show that: (i)
all pp-wave solutions supported by non-constant $H_3$ or $F_p$ fields are exact
type II superstring solutions to all orders in $\a'$; (ii) the corresponding
light-cone gauge Green-Schwarz actions are non-linear in bosons but always
quadratic in fermions, and describe UV finite 2-d theories; (iii) the pp-wave
backgrounds supported by non-constant $F_3$ field do not have, in contrast to
their $F_5$-field counterparts, ``supernumerary'' supersymmetries and thus the
associated light-cone GS actions do not possess 2-d supersymmetry. We consider
a specific example where the pp-wave $F_3$ background is parametrized by an
arbitrary holomorphic function of one complex bosonic coordinate. The
corresponding GS action has the same bosonic part, similar Yukawa terms but
twice as many interacting world-sheet fermions as the (2,2) supersymmetric
model originating from the analogous $F_5$ background. We also discuss the
structure of massless scalar vertex operators in the models related to N=2
super sine-Gordon and N=2 super Liouville theories.
