Some kind of fish tape thing....send it from the central location....plenum....and push it out to each register then attach a brush and pull it back to the central plenum ???
Ive lived in one house that had the ductwork in the slab but it was in my hvac days..... dont know exactly how they constructed it....but the jobs Ive seen around here used pvc pipe from a central plenum....and I believe you would have to pull the furnace and coil to access the plenum.
You could run a brush down each register and work it back as the fan was blowing on high......that might get some of the junk to come flying out....
maybe someone with experience will come along soon.
May Allah Guide Us Towards The Right Path and move us towards a better understanding of Allah and Allahs Creation.