  In this paper, we first obtain a general result on sufficient conditions for
tensor product modules to be simple over an arbitrary Lie algebra. We classify
simple modules with a nice property over the infinite-dimensional Heisenberg
algebra ${\H}$, and then obtain a lot of simple modules over the twisted
Heisenberg-Virasoro algebra $\LL$ from generalized oscillator representations
of $\LL$ by extending these $\H$-modules. We give the necessary and sufficient
conditions for Whittaker modules over $\LL$ (in the more general setting) to be
simple. We use the "shifting technique" to determine the necessary and
sufficient conditions for the tensor products of highest weight modules and
modules of intermediate series over $\LL$ to be simple. At last we establish
the "embedding trick" to obtain a lot more simple $\LL$-modules.
