So I figured, why not use a computer program to process and filter the audio and then tell the arduino to output each band.
I'm not following that... If the computer is processing & filtering, what's the Arduino doing?
You could probably do this with a computer and a 5.1 channel soundcard (two soundcards for stereo). Assuming you know how to do audio programming on a PC or Mac. Is it OK to dedicate a computer for this?
...outputting 5 signals simultaneously given to it over USB?
They do make 5.1 channel USB soundcards. (Watch-out for "virtual 5.1" if you buy a cheap one.)
I'm not a DSP expert, but I don't believe the Arduino by itself is fast enough for audio DSP. In any case, you'd need separate ADCs & DACs. The ADC is only 10 bits, and I don't believe it's fast enough for 20kHz audio. The PWM "analog" outputs are clearly not fast enough for quality audio.
Analog filters are probably the easiest if you are going to build something. If you need variable crossover frequencies, I believe there are voltage-controlled filter chips and digitally-controlled filter chips. You could control those with an Arduino, since you would just be sending control signals rather than doing DSP.