I'm pretty sure you misunderstood your eleceng colleague, but his recommendation to put them in series is correct. The output power of a LED is dependent on current, but the current is extremely sensitive to voltage and there is chip-to-chip process variation between them. Therefore when you put a bunch of diodes directly in parallel you are forcing them to all run at exactly the same voltage, which means (because of their manufacturing variations) they will run at (possibly extremely) different currents; in the worst case you might get all the current through just one diode, which will die, then the current will go through a different diode until that dies, etc.
Series is the best option (as per my post above) because when things are in series, they have exactly the same current through the whole chain and will run at very slightly different voltages. If you want to put several series-chains in parallel, you can, but you must control the current through each chain separately, e.g. using one of those controlled-current supplies I linked above per chain, or with a separate high power resistor for every single chain. You cannot just make 10 chains of 3 LEDs, put those chains in parallel and then one resistor on the whole lot, because that gets you back to the original problem of unequal current-sharing. You must have a separate resistor for every chain.
Resistors are simple and fairly easy, but there's a bad tradeoff between the accuracy of your current control and the amount of power wasted in the resistor. If you can possibly get them, the current-controlled supplies that are designed to directly drive series chains of LEDs are a far better option. More voltage across the resistor means less uncertainty in the current due to manufacturing variations (voltage at chosen current) in the LED chain, but also means more power lost to the resistor.
If you're going to insist on using a 19V supply, you should put 4 LEDs in series in each chain (~14V) and then use a 15 ohm 5W resistor per chain (resistor will dissipate ~1.7W). Heatsinkable resistors are easier to keep cool (they won't burn you), but non-heatsinked resistors should be used in cool, free air (half inch clearance) and not half-ass glued down to a heatsink. Air-cooled resistors are designed to get extremely hot at their rated power (100C+) but that's OK as long as their positioning is such that they can't burn people or other parts of your design.
If you put 3 LEDs in series off 19V, you need a 27 ohm 5W or 10W resistor, which will dissipate 2.7W. Clearly that is much less efficient (0.9W lost in resistor per LED) than having 4 per chain (0.4W lost in resistor per LED). And of course you need more and larger resistors.
Once you've assembled it, measure the voltage across the resistor while operating, and measure the resistor (out of the circuit), to determine the actual operating current. If the LEDs and the resistor are both at the extremes of their allowable tolerances (10-20% is not uncommon for power resistors), you might have a lot more or less current flowing than you expected from your design. In that case, you will need to replace the resistor with a different value.
