Let me try to help you since no one else has replied.
It's going to be difficult to give you definite answers because you have only supplied a PCB Layout.
Schematics would be far more useful. Of course removing the Crystal will stop the clicks, that's because once it's removed there is really no digital circuitry in operation whatsoever.
Generally Digital Pulses can infiltrate Analog Circuits very easily. So steps must be taken to isolate this.
The most common place that this can happen is through the Power Rails. The V+ Rail is just as prone to transferring them as the GND is. First each chip should have it's own Decoupling Capacitors as close to the power pins as possible. You generally want to use higher grade Caps in the range of 0.01 to 0.1µf. Ceramic Caps are the lowest grade I would use.
If you can use a second 78L05 Regulator to supply power to just the Analog Section and Filter it well that "may" completely remove the clicks. The ATmega Chip may also generate clicks when changing frequencies. Finally the coupling between the ATmega Output and the LM-358 Input may need to be AC Coupled if you now have it DC Coupled.
Hope this helps.