Yes, that would work for plain LEDs if you don't mind the 60Hz flicker. However if you want to be able to use an off-the-shelf LED strobe intended for exterior mounting or even a self-flashing LED, you'll need to smooth the half-wave rectified DC and limit it to approx. 12V.
The circuit lineup after the diodes would probably be something like a 15R, 3W wirewound resistor to limit the inrush current, charging a 50V 1000uF electrolytic capacitor, feeding a LM317 regulator with a 120R upper feedback divider resistor and a 1K2 lower resistor for approx 13.5V out, near enough the same as the float voltage of a 12V SLA battery, which most security system accessories (e.g. a LED strobe) expect as their supply voltage, and with 100uF,16V decoupling on its output. That would be good for up to 200mA to power the strobe. If it draws over 100mA the LM317 will need a small heatsink. The diodes should be min. 1A, 100V. N.B. The 1000uF capacitor *MUST have a 1/3A or greater ripple current rating for 200mA load current - if nessercery use a higher value capacitor or two in parallel to get a high enough ripple current rating. If you need more than 200mA, decrease R1 to 10R, replace the regulator with a nominal 3A Dc-Dc buck module from EBAY set to 13.5V and omit C2, R2, R3. However its not a good circuit for high load currents - if you need that you'd be better off minimally smoothing the voltage from the combined diodes and using it to switch a SSR controlling a separate 12V PSU.