Yes. IME, it's audibly a Very big deal. It's true that digital noise, for example, can certainly be transferred between components through the interconnecting wiring, but much of it is also radiated through the air and therefore physically from component to component. BTW, not only are digital sources particularly noisy, but they themselves are the most sonically susceptible to the very type of noise they radiate. But, YMMV?? The effects of such electrical noise (analog or digital) are system or listener dependent?? If I may say so, this is actually not Truly the case at all. In fact, I've come to realize from firsthand evidence that Everybody's system is Absolutely CHOKED with all manner of electrical noise. Let me put it this way: if, say, for any reason and under any circumstance, you make a change in your system in order to see if there is any audible improvement due to electrical noise reduction (you substitute a PC, try a filter or power conditioning device, move components around, or whatever) and no meaningfully audible change occurs, then it is for one (and Only one!) reason: the change you made was simply not effective enough to audibly reduce any noise, period. IOW, all of this is to say that, if you did all that, and you drew the conclusion that noise is simply not an issue, that would in truth be both drastically premature and entirely false, actually. Some products will work wonders to uncover this for you, while others little or not at all...and can even make things worse in a few cases...very much buyer beware. In fact even among audiophiles who think there may be some truth to electronic noise reduction, there's very little working knowledge of possible solutions and this is ripe for the scam artists to move in and claim whatever they want. Proceed there with all caution.
While we're at it, "dirty power" (or "contaminated") is actually is rather a misnomer. It makes us think that once power (AC or DC) has been "contaminated", then there is no way to restore it to a corrected condition without some form of filtration (as you would with drinking water). But, the fact is that electrical noise always dissipates over distance. It's like when a placid and slow-moving river with all its water moving at the same rate, regardless of depth, encounters a steeper ground gradient that's randomly lined with large rocks and boulders, creating rapids ("noise"), but then, at the end of that passage, resumes its flow as a steady, slow-moving river again. No "filtration" is needed in this case other than a sufficient distance from the rapids. If a sawmill is operating locally the AC will be contaminated, but beyond about a 5-mile radius things will have returned to what we'd call normal. The larger the current and the higher the voltage (and also the greater the noise), then the greater the distance required to return electron transfer states to normal. Lowrider is correct in that electronic noise is bi-directional - it moves upstream as well as down. Additionally, noise reduction, for example, has been loosely associated with dedicated system Romex lines from the circuit breaker box. Dedicated lines themselves actually cannot reduce any noise except by physically locating (by way of a longer connection to the box) the system from the rest of the noise in the home. But, my point in this paragraph is really that if you see a manufacturer specifically using the terminology of "dirty" or "contaminated" power and use them as if they were validated terms, that should be your first red flag. They have picked up these terms from audiophiles in forums who are still struggling with noise reduction concepts and are plainly out to use their ignorance against them.
Alright, so what causes the noise? Every circuit on earth vibrates, the exact frequency of which is dependent on a number of things. But, they do vibrate. Like that occasional AC, utility-pole transformer that you can hear humming from more than 50 yards away...or the hum you can feel on your preamp's transformer. Some circuits will seem to hum more than others and some will be vibrating on a level too weak for us to detect. In audio equipment the vibration compromises the performance of all the parts (resistors, caps, diodes - everything). IOW the noise is necessarily self generating. Not only is that true for all our components, but in fact every appliance or electronic device that is plugged into your home. Every coffeemaker, alarm clock, cellphone charger, wifi router...everything. Remember, the noise tends to dissipate over distance, so things on circuits on the opposite side of your home from the breaker box leak less noise back to the box (where it is from there redistributed all over the house and, in the process, back to your components [the box is a kind of grand central station for your noise sources]).
Very few audiophiles may have firsthand reason to realize it at present, but I will say that electronic noise is not merely "a problem" for us. IME, it is THE Single, Largest Obstacle for realistic sound (and video) reproduction in this hobby and in this industry, period - regardless of what Anyone paid for their system. The potential gains to be had out of this approach are in fact enormous. It is an exceedingly difficult thing to overstate. I say that not so much out of enthusiasm (which I have), but out of the insights I've managed to gain from the sonic results I've achieved with it, over the last 5 years.
As with Lowrider's post above, the immensity of this field of study is limitless really, and I, too, am not truly any expert. It has taken me a while, but the cumulative effects of the firsthand experiences I've had over the last few years have at least allowed me to finally "get it".