Can I use an LDO to control several diodes at a speed in the order of kHz?
Possibly.  The datasheet gives us some useful info:

First, from page 3, "This pin (out) requires an output capacitor to maintain stability".  The output capacitor is required to be at least 22uF.

Let's say your LEDs draw 20 mA each, or 60 mA total.  I = C * dV/dT means that if the regulator shuts off instantly, the voltage will only drop by 0.27V in 0.1mS (1/10th of your period).  Exactly how long it takes between shutting the regulator off and when the LEDs turn off depends on your exact circuit, and on how much current the LEDs are drawing.

The turn-on transition will have a similar delay, because the LDO needs to have time to charge that capacitor.  There aren't any turn-on waveforms in the datasheet to reference, but at 1A it will take about 0.04mS to charge that cap, which is probably OK.