3ms Camera LED Flash
Note: Some of the information below is anecdotal, since my measurements use a Canon 1D Mark III dSLR, rather than specific light-sensing instrumentation

I've just started work on a strobe softbox for my own product photography studio use, especially motion capture (multiple exposure of moving parts in a single frame): I will be using RGB LED strips to get color control.

Couple of things to note: 


If you have 12 Volt or 24 Volt LED strips, those already incorporate the current limiting resistors in each cut-able piece, so you do not need current limiting - Just provide the rated voltage. 
A 3 millisecond strobe every 500 ms is trivial to achieve with most LED power supplies and LED strips: I have been experimenting with a 40-Watt LED power supply, using MOSFETs to switch power to the R, G and B lines, and I am able to achieve strobe duration reliably down to 2 strobes in 1/8000 second (0.125 milliseconds).


My results get muddy below around 30 microseconds per strobe. My camera does not support a shorter shutter time than 1/8000, so I cannot capture more precise data.

Lessons learnt: 


A 24 Watt power supply used on a 24 Watt LED strip leads to spurious pulses during the time my shutter is open, but a supply with at least double the LED strip's rated power works fine.
Minimum clean strobe duration varies widely between LED strips of different manufacturers, and slightly between different LED strips from a single manufacturer.
Pulse edges get softer due to capacitance along the conductors, as the length of the LED strip increases, so connecting lots of the "cuttable" cut pieces in parallel works better.
The green intensity needs to be reduced a lot compared to the other two, the red a bit lower than the blue, else a color cast appears.
The green needs to have duration reduced by about 0.25-0.5 microseconds compared to the other two colors, which may be due to secondary phosphor afterglow: See this answer for a detailed discussion on the "long tail" subject.


The last two points are not applicable if white LEDs are used. However, white LEDs will have a slight color cast when using very short duration (say less than 0.1 millisecond, exact limit depends on specific LED used) strobes, due to the yellow-emitting secondary phosphor continuing to glow after the blue / UV junction emission stops at each pulse. At 3 millisecond strobes this color cast will not be discernible. 

Recommendation: 

Experiment with the specific LED strip you intend to use, to determine the point at which any shorter strobe duration causes a color cast. The figures should be far shorter than the 3ms specified in the question.