Why does my MSB DAC III click?

Do you have the manual?I believe your problem relates to a user adjustable jumper.If you need specifics,go to the MSB website and download the manual.

I looked at my manual.If you look at the main board inside your Link,I believe you will find that the jumper has been removed from position 1 on IC J14.This controls the switching of the DAC to analogue bypass.When the jumper is in place the DAC is active as long as a device is connected and turned on.With the jumper off,an active digital signal must be present to "keep" the DAC on.

That was very kind of you to look that up Birdieman. Kind of cool that the MSB actually has user accessible jumpers to adapt to / correct situations like that too. Sean
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I'm also experiencing the same thing with my DAC III. I leave it on all the time but when I turn on my cd player which I use as a transport, it clicks. Same thing when I turn the cd off or when I unplug the dac. Similarly, I hear it with my cd player (rotel rcd 970) when it is turned on.
I just thought that this was normal because the relays are switching something on. Birdieman's explanation made things clearer for me too.

I'm wondering whether moving the jumpers could also affect/improve the sound.

Cut your fingernails? The increase in PRaT from this fine DAC simply has you tapping away! Take your shoes off, too!
(Sorry...couldn't resist.)
Have a fine Thanksgiving, all.