  We study the effect of dissipation on the infinite randomness fixed point and
the Griffiths-McCoy singularities of random transverse Ising systems in chains,
ladders and in two-dimensions. A strong disorder renormalization group scheme
is presented that allows the computation of the finite temperature behavior of
the magnetic susceptibility and the spin specific heat. In the case of Ohmic
dissipation the susceptibility displays a crossover from Griffiths-McCoy
behavior (with a continuously varying dynamical exponent) to classical Curie
behavior at some temperature $T^*$. The specific heat displays Griffiths-McCoy
singularities over the whole temperature range. For super-Ohmic dissipation we
find an infinite randomness fixed point within the same universality class as
the transverse Ising system without dissipation. In this case the phase diagram
and the parameter dependence of the dynamical exponent in the Griffiths-McCoy
phase can be determined analytically.
