Hi there,
To add to the previous posts a little here...
Switching power supplies use rectangular waves to convert energy from one voltage level to another. Simple cheap wall warts do not do this.
Rectangular waves generate harmonics way up there in frequency which can propagate just like a radio wave. Sensitive circuits pick this RF up as noise and this could happen even with the circuit not connected. An AM radio for example may pick up this noise on one or more channels and that could block the channel due to the noise level being higher than the signal to be detected.
In addition to that problem, they also produce a small amount of ripple at the switching frequency. It's not too hard to get rid of this however, as a balanced inductor capacitor filter can get rid of that noise to a great degree. One idea then might be to use two inductors and a capacitor or two to filter the DC so it provides a cleaner voltage to the circuit. Even small hand wound coils can help knock the noise down quite a bit.
For the problem of RF noise, you can try moving the power supply farther away from the circuit itself, but you'll also need the balanced filter on the output. If that is not practical, you could experiment with shielding the circuit inside a metal can or box. Another idea would be to design the circuit so that it is less sensitive to noise, which can come as easy as reducing the impedance of some of the more sensitive circuits sometimes.