A common trick is to connect a diode across (in parallel with) R such that it is reverse biased in normal operation. For example: if V is the positive supply in your schematic, connect the anode to Vc and the cathode to V.
When the supply voltage is removed, the diode becomes forward biased and the capacitor C rapidly discharges via the diode [as it attempts to drive the supply and power the rest of the circuit].
The traps to consider with this simple approach are:
* C only rapidly discharges down to the forward voltage of the diode (~0.6V for silicon, 0.3V for schottky)
* The reverse leakage current through the diode D can swamp the current through R - especially if R is big, the temperature is high or D is a schottky.