Control many RGBs by color group using transistors instead of using LED Driver ICs
For common cathode RGB LEDs, high side switches (P-channel MOSFETs) will drive the color channels, while low side switches (N-channel MOSFETs) will switch each LED on or off.

See the schematic below.







simulate this circuit &ndash; Schematic created using CircuitLab

For the P channel, FDN340P is available for less than 8 cents each in 50 unit lots.

For the N Channel, IRLML2502 is available for less than 10 cents each in 50 unit lots.



Now for the complications with such a solution:


LED perceived intensity differs between colors. Human vision is most sensitive to yellow-green (~555 nm wavelength), and pretty low for blue. See the "Wavelength" heading of this answer.

So different drive intensities would be needed to make the 3 colors relatively similar visually. This can be achieved by experimenting with different resistor values, to adjust the current through each color of LED, from the nominal ~20 mA used for the resistors shown in the schematic. Often, 10 mA through green looks similar to 20 mA through blue, but this varies from LED to LED and between batches. Many integrated LED drivers have adjustments (grey / dot correction) available for this.
Perceived intensity delta with applied power (in this case PWM duty cycle) is non-linear. In other words, by linearly increasing duty cycle, perceived intensity will not vary linearly. An exponential function of around x^2.5 is recommended for perceived linear intensity increase. See this answer for an explanation.
Real-world discrete MOSFETs have significant variation in Rds(on) between units and with temperature. While this may apparently be irrelevant for a switching application, in reality with the very small voltage headroom available with a 5 Volt supply, even minor resistance variations will cause the voltage across each MOSFET to differ under conduction, and this will affect LED intensity. Integrated LED drivers address this through grey adjustment or dot adjustment, as well as on-chip trimming between drive channels. All channels sharing a substrate also ensures that the driving FETs are more or less equal in temperature, even if some channels are momentarily off while others are on. 
To drive the proposed arrangement, 12 + 3 = 15 pins will be needed from the microcontroller, as opposed to 2 to 4 pins for driving a serial (SPI or I2C) LED driver.
The cost of the required number of MOSFETs, and the size of PCB involved, may well exceed the cost of a single suitable LED driver IC and its PCB.


In short, while this approach may sound viable in theory, it basically solves one problem by introducing several others. Not practical from an engineering standpoint.