Hi Jack,
In the last 6 years that I have played with the AIC3106 in the lab, I have not had any problem duplicating datasheet performance. This is a very high volume CODEC used by many, many customers with great success, so I suspect there could be an issue with your test setup.
some areas to investigate would be the clocking and digital audio interface (it looks like you are in slave mode which puts a lot of the clocking burden on the host), and the analog input/output range (avoid clipping).
In the register settings you sent, there is no DAC path shown for register 7. I suspect this was a register dump vs. an I2C capture at the I2C pins as we had requested. The sequence of register writes can also be an issue.
Do you have any shots of your clocking (MCLK, WCLK)??
Do you have an AIC3106 EVM? This would rule out hardware problems on OMAP board.