As a long term user of these devices, I believe that they "only" respond to the points at which the cuff pressure overcomes the heart output when it is in systole and diastole. The MD or GP does actually listen with a sphygnomanometer (stethoscope) to the points where as the cuff pressure drops the sounds that Jimbo referred to are where the the cuff overcomes the pressure in the arteries and no blood flow is heard. There is one loss of sound when the diastolic pressure (the lowest value, normally 80-mm mercury) is overcome and the other when the systolic pressure (the highest value, normally 120-mm mercury) is overcome.
You need to do some sums to convert mm mercury into bars and atmosheres to get the units right - you might need a gauge or absolute sensor rather than a differential sensor.
All the commercial BP monitors work with inflatable cuffs that are pumped up to a pressure in excess of the systolic pressure (also sphygnomanometer ) then the pressure is allowed to drop while the sounds are "listened" to. The tricky bit is doing it all with air pressure measurements (look up oscillometric method) where I guess the pressure sensor can pick up the subtle changes at diastole and systole. You might be able to find a way of using a microphone to pick up the arterial blood flow noises
It might be worth looking at some of Libellium's (Cooking Hacks) life science sensors and I believe that Texas Instruments are quite strong in this field.
Good luck and mind your BP.