Allegro makes a 16 bit constant current latched LED driver.
I'm not sure the full current rating it would support, but it essentially removes the need for the current limiting resistors, you use on resistor to set the current you want to limit it to and it limits the current on each pin to that value.
you couldn't do it in series because you have only one cathode for all 3 colors. if it were in series would essentially be connecting all three anodes to the cathode of the prior led, thus making it an expensive white LED. for 3 RGB in parallel, you'd connect the cathodes, and connect the anodes for each color separately. in other words three sets of 3 anodes you'd need to adjust the current you put out because putting them in parallel draws more current (60mA vs. 20mA)
how many you need to control at any one time. take a look at the hypnocube, they don't use the constant current driver, but rather the 74hc595 and separate resistors, but the concept remains the same, they have 4 cathodes for 16 RGB LED's (one for each layer of 4), and 12 anodes (4 columns of each color) this is repeated 4 times to make essentially 4- 4x4 matrices. it varies the color by sucessively pulsing the three different anodes of each LED at different frequencies to make it appear the colors between red green and blue. As for the code, it would be pretty long because for each apparent change of color, you'd need to put in numerous instances of the same sequence to vary the colors. it is probably much less work to figure out varied pattern with a math function rather than a data table that would take a lot of data entry for a relatively short pattern. you can increase the hold time for each data set, but the more you do that the more flicker will be noticeable. there was a thread by retrolefty, for an 8 cube of single color, or 512 led's; if they were RGB it would be essentially 1536 led's. your 99 LED's is really 297 LED's if you want to be able to control each LED independantly. if you control more than one per out put you can cut this down. you were saying three at a time. I'd look at the datasheet for the drivers to see how much current it can handle per output.
one risk you run doing three in parallel, is if one of the 3 fries, the other 2 will likely fry not long after because the current will still be allowed to flow as if there were 3 there and thus overloading the other 2. this is rare but not unheard of, especially if overheated during soldering.
If you are going for the same color for all the led's at one time the FET would work, but from what I understand you want more than just a 7 color (red, green, blue and combinations thereof) if you do just want the basic colors, it would be best to do them individually, with current limiting resistors for each color of each led. I guess I'd need a better Idea what effect you are trying to accomplish. RGB LED's aren't cheap, and to make the most of them I suggest playing around on a small scale before going full scale.
I hope this doesn't overwhelm you, there is unlimited potential with the arduino, I just don't want you to get discouraged by biting off too much on your first project with something that can become costly quickly.