To a certain degree, mind-body duality has persisted even in the absence of a belief in a "soul" -- it's just that we think of "body" and "brain" as separate entities, as if the brain is just piloting a meat-vessel around.
The key thing here is that feeling bad is completely inseparable from the physiological mechanisms that underlie it. At a very high, very superficial level, the stress mechanism is something like this:
The amygdala, which is responsible for integrating sensory perception with memory and emotional processing, determines when a situation is "stressful." The amygdala then signals the hypothalamus via the amygdalofugal pathway and stria terminalis. The hypothalamus secretes a number of hormones which basically prepare the body for fight-or-flight behavior. Of particular interest is corticotropin-releasing hormone (CRH) which, along with vasopressin, activates the hypothalamic-pituitary-adrenal, or HPA, axis -- the physiological mechanism of fight-or-flight.
CRH is responsible for stimulating the pituitary to release cortisol, which is an absolute sledgehammer of a hormone: it stimulates glucose synthesis (NEED MOAR ENERGY NAO!), regulates the sodium-potassium balance in an effort to stave off hyperkalemia (probably an adaptation to deal with tissue damage), reduces inflammation (through immunosuppression), constricts blood vessels, suppresses the parasympathetic nervous system (shutting down digestion -- not important when you're trying to avoid being eaten), and so on. These activities are the result of cortisol binding to glucocorticoid and mineralocorticoid receptors; such receptors are found in all kinds of tissues.
Cortisol also overwhelms the cortisol receptors of the hippocampus, which processes memory. The hippocampus appears to be part of the downregulatory subsystem of HPA axis homeostasis, but if its receptors are flooded, it becomes ineffective at suppressing HPA axis activities. This results in both the impairment of the memory-processing functions of the hippocampus, and a reduction in the negative-feedback mechanism of the HPA axis. In other words, you can't stop stressing, your comprehension is impaired, and your body is going into overdrive.
HPA axis stimulation is useful when stress is a rare event tied to things with sharp teeth; it's counterproductive when applied to our modern world, when stress is more likely to come from deadlines, traffic, and trying to find that last parking space at the mall on Black Friday.