You CAN run LEDs off a suitably chosen regulated voltage without a resistor, but you have to be careful since rather small changes in voltage can destroy the LEDs, and choosing the voltage can be tough since the datasheet numbers are "typical" and do vary. If you have a grab bag of LEDs, you won't be able to get them all at the same brightness, either. If you don't care about any of that, then yeah, have at.
The reason for the resistor is to attempt to overcome the nature of LEDs that very small fluctuations in voltage can result in a large change in current and therefore power which could destroy the LED. LEDs aren't nearly as forgiving of voltage issues as, say, incandescent light bulbs. In fact, they can be so intolerant of it that it can be impossible to get LEDs of the same lot to light at equal brightness with a constant voltage source. The resistor swamps this intolerance (it's an exponential dependency of current on voltage) with the "well behaved" nature of the resistor (which is a linear dependency of current on voltage). If you have different color LEDs, then, at minimum, each color will need a slightly different voltage.
If you have a bunch of LEDs and don't want individual control, you can wire them in series using a single resistor. This is how long strings of LEDs for lighting purposes are often made. Choosing the resistor isn't a terribly complicated process. Just add up the expected voltages on the LEDs, choose the lowest current that they can all handle (this does NOT add as the current runs serially through each LED), and choose R = (Vs-Vl)/I where Vs is your supply voltage, Vl is the added up LED voltages, and I is the expected current in amps to yield a resistance value in ohms (remember that 1mA = 0.001A). Vs-Vl must of course be positive and should be at least 1.5, so you may need to use a 12V or 15V supply instead of a 5V. Note that the power in this resistor can become somewhat substantial if you use an inordinately high voltage. P=I^2*R (same definitions for the variables as before). You can buy a 1/2W or even 1W resistor if you need to. The LED-Wiz is probably good to a 25-30V supply, but check the manual before using anything over 12.
Of course, you can have several strings each with its own resistor. This is one way you get 500 LEDs on 32 outputs from the LED-Wiz, in fact. You'd only need 32 resistors, then.