How important is the amp?

In the sense that an amplifier is a fundamental and necessary component to any audio system, they are important. As far as it's contribution to the quality of the sonic signature the system produces, provided it is functioning properly and is of adequate power to properly drive the speakers attached to it....not so important. I would think components' affect on sound quality ranks something like this:
Room Acoustics
Loudspeakers
Recording Quality
Preamp
Source Hardware (TT, CD, Computer Configuration, etc.)
Power Amp
Wires and Cables.

How important is the amp?

Everything is important.

No amp, no sound.

Good amp up to the task of driving speakers to their max, great sound.

Then there is everything else in between.

So it's pretty important I would say.

10-09-15: Danoroo
In the sense that an amplifier is a fundamental and necessary component to any audio system, they are important. As far as it's contribution to the quality of the sonic signature the system produces, provided it is functioning properly and is of adequate power to properly drive the speakers attached to it....not so important. I would think components' affect on sound quality ranks something like this:

sorry but i strongly disagree.
do not underestimate the amp-speaker electrical interface. many people here on Audiogon (& other places) have gotten themselves into loads of trouble both electrically & sonically by blowing this off.
The amp is a major part. You get this interface wrong & all your hard work for the components upstream is flushed down the toilet.

Like it or not, the amp is the heart of the system. It's absolutely not the pre amp. Only one of these can be eliminated without consequence. I find the synergy factor laughable except for certain speakers requiring specific characteristics. I've owned lots of speakers and amps and have always been able to identify the amp regardless of whatever else is in the system. So you always end up listening to your amp no matter what other changes you've made. I challenge anyone to not recognize the sound of their amp through any and all changes they've made. It's the only component in the chain that this can be attributed to.