multiple RGB LEDs with common resistors for each color
Series resistors for LEDs are used partly because LEDs have a negative temperature coefficient, and this leads to thermal runaway.  When the LED conducts, a portion of the energy passing through it is wasted as heat, proportional to the current flowing through it.  If the LED is not able to shed heat quickly enough, it heats up, which decreases it's resistance.  This increases current, and in turn increases the temperature of the LED.  

A series resistor has a positive temperature coefficient, so as current through the resistor increases and the resistor heats up, it's resistance increases, decreasing current again.  In order to accomplish this, there is a minimum amount of power that the resistor must waste for it's temperature coefficient to counter the LED's, and even in commercial products like flashlights or headlamps, the resistor can be using almost half the power of the LED, which bothers me when I see it.  At lower current levels, because power losses due to resistance are \$I^2R\$ this can be negligible, but for power LEDs it adds up to a lot.

The problem with having multiple LEDs share a resistor is that once current passes through the resistor it will be shared between the LEDs, depending on their individual properties.  This will almost certainly result in one LED drawing more current than the others, which will heat it up and cause it to draw more current yet.  The resistor will be limiting the total current, but not forcing the LEDs to share it properly.

One other purpose these resistors can have is to be sized (in wattage) to act as a fuse for that LED.  A resistor is chosen of a type that will fail open (which I believe most if not all do).

This is not the only way to prevent thermal runaway, only the simplest.  To remove the resistor from the circuit and still have the LED work properly, you have to accomplish the same tasks the resistor is performing for you.

1.) Regulate voltage - LED efficiency tends to respond to instantaneous voltage, so while you can dim or decrease current through an LED with pulse width modulation, the LED's efficiency will tend to suffer if the voltage being pulsed to it is too high, even if the LED doesn't burn out.  This will matter less the higher the frequency, as parasitic inductance starts to smooth things out, but I found at around 200 to 1000Hz, if I PWM 10V to bring the LED to the same brightness as giving it a constant 3.2V, it gets noticeably hotter.
You can regulate voltage many ways, including selecting a different power source, but depending on how much voltage drop the resistor is giving you, it may be well worth regulating the Pre-PWM voltage to the LED.

2.) Prevent thermal runaway - regulating voltage over an LED is only sufficient if the LED is being cooled at a rate better than a certain threshold.  A series resistor is regulating not just the voltage applied to the LED, but also the current through it, so this can be accomplished by using current mode control of the LED.  A 100mA current mode driver will give the LED the same current regardless of how it's resistance changes, so as the LED heats up, it will reach a thermal equilibrium and rather than current changing due to temperature, brightness will subtly decrease, but a runaway condition will not occur.

3.) If you're making a commercial device with a warranty, it's much cheaper to repair a 2c resistor than a 30c LED.  In one case as an electrician I made this precise repair of some LED pucks.  In many cases devices are not worth repairing, but even in the modern world they can be if the repair is simple enough not to necessitate removal, replacement, reshipping, keeping a customer waiting, ETC.  So if this is the case it won't hurt you to size the resistors to act as fuses.  If you don't want to use resistors though, this function can be performed by a protection circuit on the switching driver.