
The first step is to make sure that there is no DC component, intentional
or not, on the audio lines to be switched. If neither the output or
input presents an unintentional DC level (let's say they're capacitively
coupled), then putting a 22 K resistor from each input to ground, and
from the output to ground is a good precaution against leakage in any
coupling caps presenting a DC level that is GUARANTEED to cause a click.

If the device presents an unintentional DC level that is from more than
just coupling capacitor leakage, then you need to remove the DC before
switching with coupling capacitors of your own.

If you have more than one relay (each relay switching one input
through to a common output), you can use make-before-break switching
where for a brief instant both sources are connected during a switch
change. This may mask the switch click.

An audio switch that is clickless is made by taking a light bulb and
making your own ersatz opto-isolator with a light-dependent resistor (LDR)
sealed inside a piece of black heatshrink. Many broadcast consoles
use this older, but time-honored technique. Claro actually makes
a complete optisolator like this, but I don't have a part number handy.
Radio Shack has a LDR grab-bag (276-1657) you might want to try.

For a simple, solid state clickless switch that is cheap, you might try 
looking at the CD4066 analog switch that is commonly available (or its
older predecessor, the 4016). Again, the DC component must be removed
prior to switching. Other alternatives include JFET switches (both discrete
and IC) and diode switching (which works relatively well, surprisingly).
All the above, with their pitfalls and application tips could easily
make a subject for a multi-page article.

Solid state switches may induce a click of their own when turned on
due to charge being injected into the audio line through device
capacitance when the control signal changes state. Techniques to
minimize this include lowering the impedance of the signal paths through
the switch, differential switching, and slowing the risetime of the
control pulse. With the 4016, cutting the click can be as simple as
putting 10K - 47K resistors in series with the control pins.

Hope this helps at least get you started.

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