What could be causing these mold issues in my brick townhouse?
DA01's theory is correct. Outside humidity is migrating through the wall until it contacts material colder than the dew point of the vapor laden air. The vapor at that point condenses into liquid water. Aside from not using A/C, one remedy is to install a vapor barrier between the insulation and the exterior wall cinder block. This has a detrimental effect in the winter because the situation reverses. You need a vapor barrier on the interior side of the insulation. This is feasible, but the problem now is any moisture that gets past the barrier (no barrier is perfect) has no where to go. It is trapped in the insulation.

This of course means removing all insulation and drywall on exterior walls. Thus, I believe the best course of action is to insulate with a material unaffected by moisture and is actually a vapor barrier itself. One such insulation is spray applied polyurethane. The only drawback of this material is that it is quite expensive. It's the ideal material for difficult, varying climatic conditions such as yours.