If your LED's are common anode, then you would require a way to provide a 'high' level voltage (+12V if thats your supply) when your PWM controller outputs a low. This means you need a PNP transistor (PNP for high side, NPN would be used for low side). Or, for MOSFET's, a P channel.
Sounds like your PWM controller has built in drivers for the LED's. As their output can be up to 3A, these could easily pull a P channels gate low, very quickly, but as I imaginge the output of the PWM is open collector, the MOSFET's gate would need to be drive high when the PWM is off, probably by a resistor.
I would recommend that, provided you can't replace the LED string with common cathode, that you use three P-channel MOSFET's, designed to switch > 5A. Ones that are designed for high speed would make it more efficient (you're looking for low gate capacitance, and low switch on/off times).
What LED's are you using? If they use less than 1Amp, it would be easier and cheaper for you to use PNP transistors instead of mosfets. Eaiser to drive, cheaper, and generally nicer to work with, but not as efficient for switching high current as a P channel MOSFET.