By directional, I mean that the sensor is only receptive to sound coming from a given direction. Microphones of all sorts have charts describing their directional sensitivity.
If you rotate your whole array, the directional limitation can actually be an advantage. It can help you eliminate secondary echoes.
With omni directional mics, you wouldn't need to rotate the array. Your transmitter would still need to rotate so that the echoes would have a known angle of incident. An omni directional transmitter would not give you the result you are looking for.
I would place a transmitter with a rotating reflector in the center of an equilateral trangle formed be the three fixed, omni directional mics. One way to make the three mics omni directional is with cone shaped reflectors.
This is still only a two dimensional result. To get the third dimension will take some doing. The transmitter would have to rotate on two axis. Or you can use an additional receiver to form an equilateral, triangular pyramid. Or you can orient your three receivers on the three axis, rather than on a plane. That would still require omni directional mics or rotation.