I've tried powering the board using both the USB and a 9V battery and using a star topology for ground, but the noise is still present.  I haven't tried decoupling each module's power yet, but I'll try that next.

What you've explained makes a lot of sense.  I used the SPI header to connect to a breakout board and it works just fine.  Thanks!

Reads on an SD card can be heard through the Dac.

Arduino Due
AdaFruit Micro-SD Breakout Board (Board -> SPI and Pin 10)
AdaFruit MAX98306 Amplifier (Board -> DAC0, GND, 3.3V)

Things I've tried to get rid of the noise.
1.  Redid wiring using Jameco connectors -> still noise.
2.  Bypass Amplifier to externally amplified speakers -> still noise.
3.  Tried Dac0 and Dac1 -> noise on both.

I've attached a picture of the wiring.

Any help is greatly appreciated.

I just bought a Due board and am really impressed.  I'm very new to electronics and I'm still trying to understand all of the 3.3V and 5V differences and ramifications.

// Pinout diagram
http://arduino.cc/forum/index.php?topic=132130.0

I'm a little confused about the SPI header.  If the MOSI, MISO, and SCK pins can only handle 3.3V why does the SPI supply 5V?

I ordered a Due today, and decided to take a look at the IDE.  I downloaded the 1.5.2 version onto a fresh XP machine along with JDK 7.  The IDE crashes after showing the splash screen.  All other crash reports were on version 1.5.1 or 1.5.1r2 and don't resolve this problem.

Are there any log files that might point me to the problem?