How do cockroaches resist the effects of ionizing radiation?
Off the top of head as a medical professional I can imagine the following mechanisms (everything is just speculative reasoning):


Insects don't have blood. Instead, they have hemolymph whose primary role is not oxygen transport (they have an additional tracheal system for this purpose), but rather that of nutrients. Thus they don't need (and don't have) an intense proliferation of blood cell precusors -- these (bone marrow, spleen) are the most susceptible to radiation in a human and animal body.
Insects have a rather primitive immune system that is mostly humoral[a]  and much less cellular[b] compared to the immune system of animals and humans. This eliminates the next common weak place in the body: lymphatic nodes, thymus, again spleen and bone marrow etc.
Insects have generally a much primitive and in many cases also rather decentralized nervous system: the ganglia are organized in a sort of a cord and even though the capital ganglia are usually larger, these dominance is not as prominent as in case of CNS and PNS in animals and humans. Therefore this system is much more tolerant to losses.


1.-3. Therefore, the only sensitive part of insects is the intestinal epithelium which gets renewed on a regular basis (similar to that of humans, also a known target of radiation), but...


Insects (and generally the arthropodes) are known to have exoskeleton. This potentially serves as a good "armor" for vulnerable intestine cells, filtering out the most heavy particles (like alpha- and in some respect also the beta-particles). 
EDIT: This seems not to be real protection, see the discussion in comments.


Therefore it is not a surprise that insects generally show much higher resistance against radiation.

EDIT:
As it was correctly added in the comments, there are also gamets, that are most sensitive to radiation (because they bear only the half of the normal genetic information and cannot repair mutations). Even though the lesions in gamets do not lead to immediate death, the potential sterility can easily cause the extinction.

However, cockroaches (and insects generally) are known to be r-animals, meaning that they favor the quantity (r) over quality (K) of their off-spring. This strategy is optimal when dealing with radiation-induced changes in gametes: the high number of offsprings compensates for the genetic imperfections in gametes.



[a] -- meaning that is has secreted peptides in their hemolymph that protect them
[b] -- there are phagocytes, somewhat similar to tissue magrophages in humans, but the rest of the cell chains in immune response in vertrebrates, like T- and B-cells, are completely missing. Those are responsible for the mediation and amplification of the immune response in vertebrates and are the cells that are most susceptible to radiation damage.