I don't know if you would consider this idea a simplification but,
you could use a triple 3-input OR gate IC instead of the diodes.
That might work. I have an idea that uses a lot of diodes but if I could get something like what you suggested that definitely might make it more simple. I'll come back and let you know if what I was thinking of is feasible.
Rapid80:
Lol, that is a perfect-one liner. I don't know why but that cracked me up. Anyways, I completely understand that technically this is the wrong place to ask, but I have gotten a ton of help here before and I know that you guys are skilled electronic geniuses, so I figured I would ask. I greatly appreciate any help that is given, so thanks so much even if this is the wrong place to ask.
This is actually the right place for general electronics, but that was too good a straight line to not respond.
I completely agree. That was a good one. 
I was thinking of a way to use diodes, but it would require many that it doesn't simplify the circuit. I'm kinda stumped, maybe I can't really simplify it.
From your post it is not evident that you are aware that different color leds have different forward voltages
so you can't use the same current limiting resistor for all leds. Why ? If the Vcc is 5V and the led is a red led
with a forward voltage of 2.2V and you want to deliver 10mA to it then the current limiting resistor would be (5V-2.2V)/0.010A= 280 ohms. Ultra Super Blue, on the other hand , has a forward voltage of 3.5 to 4V, so , if , for
example, you used a "one size fits all" approach , with a 280 ohm resistor, it would only get 0.00428A (4.3mA),
so of course it would be dim. When designing a circuit with different colored leds, you need to first learn
what the color means with respect to how you design the circuit. In short, you need to look up the forward
voltage for every color and calculate the current limiting resistors for each color. That's LEDS 101.
In short, do your homework first.
The person you should be talking to is CROSSROADS. He sells display driver boards that would probably make
your project easier.
On the other hand, green LEDs are more visible and efficiencies differ, so the different currents often do not matter at all. 
I am using a different resistor for every led and none of them share one. Also, I did some tests and the resistor values that I used were the right ones to make the pink and orange colors. I was confused why the blue LED was dim because it has its own resistor so the different forward voltages shouldn't matter.
I am using a different resistor for every led and none of them share one. Also, I did some tests and the resistor values that I used were the right ones to make the pink and orange colors. I was confused why the blue LED was dim because it has its own resistor so the different forward voltages shouldn't matter.
If you reread my post, the point is not that you need a different resistor for each led , the point is the VALUE needs
to different for each COLOR.
R10, R14, & R19 SHOULD BE 120 to 220 ohms, NOT 1000 ohms because they are BLUE.
You should recalculate for the other colors too.
I used the resistors I did because they make the correct color, so how could I keep that color but change the resistor values?
I haven't thought of anything I could do, so should I just keep the circuit as it is? I still don't know why that blue LED is dim, though.
Reread post #26
I'm a little bit confused because I am using the same setup for the other blue LEDs but they are not dim. Also, I purposely used the resistors that I did to make the right color. The red LEDs use a 100r resistor to be bright, but the blue and green LEDs use 1k resistors to make the correct color (pink or orange). The resistors were chosen on purpose to make the right color, and all of the LEDs are lighting up properly except for that one.
What if replace the resistors and LEDs with those on other properly lit lines?
Try three patterns that replace both the resistor only and the LED only.
I played around with the breadboard a little and got it to make the right color. I think the breadboard was adding too much resistance, or there wasn't a good connection.
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