What resistor to use with this RGB LED?

You should use 3 different resistors, one for each color, although the blue and green have the same specs.

At 150mA the forward voltage for the red is 2.2v, green is 3.5v and blue is 3.5v. So you should use a 22ohm 1watt resistor for the red, and 10ohm .5watt resistor for the green and blue. You have a bit of wiggle room on these figures, and if you don't have a resistors that can handle that wattage you can use more than one in parallel just make sure you calculate the correct resistance between them.

Also I doubt your microcontroller can provide 150mA (it's probably more like 20mA,) so you will probably need to use a transistor on each color so that they can pull enough power. Take a look at this image for how to hook up the transistor to your system. Although ignore the 12v and multiple LEDs.

You may also want to have each color driven by a PWM pin, so that you can alter the brightness of each color to change the overall color at will.

In a rush to get out of here, but hope this helps. Ask question and me or someone will answer :)

You absolutely need a resistor, in fact, you need three. The listed maximum DC current is 150mA. I would not want to push it that hard. Forward voltages come from the table you listed

R: (5V-2.2V)/100mA = 28 Ohms

G: (5V-3.5V)/100mA = 15 Ohms

B: (5V-3.5V)/100mA = 15 Ohms

Those resistors are there for current limiting. If they are not there, your LED will not last long. Also, 100mA is way more current than your microcontroller will be able to source. You'll need to add a switch to each LED in the package.