The biggest problem with your solution is going to be as you said, the high heat dissipation on the resistors. Instead of burning up all that heat, try building a constant current source.
I had an project with some high power LEDs as well.
After some googling, I decided that this constant current source was easily the best option for me powering high current LEDs from an arduino.
Might not be the absolute cheapest option (see ebay for LED drivers), but the simplest I could find that I could build and play around with:
Calculated the setting resistor value and built x3 channels. At 350ma per channel (same as your LED), the thing works a treat 24/7 and the mosfets on a heatsink put out barely any heat.