Dazzdax, if you want the amp to not impart an electronic quality, I can tell you that the list of candidates is not very long. On top of that, its likely that you will have to accommodate the amplifier in some way, as the relationship between any amplifier and speaker is paramount.
First, in order to avoid an obvious electronic quality, the amplifier has to be designed with intention to obey the rules of human hearing, rather than the more arbitrary rules that are commonly used to get the best bench measurements.
For example, we all know that humans have bandwidth of about 20Hz to 20Khz. To accomplish that without phase response issues, the amplifier has to be good from about 2Hz to 100KHz or so. Otherwise it will impose soundstage issues and tonal issues near the frequency extremes.
For audiophiles, one of the more important 'electronic' issues is that of brightness and other high frequency artifacts. The only way to avoid this is to intentionally design the amplifier to not make the odd-ordered harmonics that the human ear uses as loudness cues (and so is sensitive to 100ths of a percent!).
To do this you cannot employ loop negative feedback as it is known to enhance those very harmonics that the ear uses to gauge loudness.
So now we are limited to amps without NFB. How do we achieve linearity? Distortion **has** to be kept down, even the even-orders that our ears do not object to, because another human hearing rule, masking (wherein a louder sound will mask a quieter sound), means that the non-objectionable distortions will nonetheless mask low level detail.
To keep distortion down you have to use every trick in the book: Class A operation, the most linear devices (more on that later), simplistic circuitry (by 'simplistic' I do not mean crude or primitive BTW, which includes but is not limited to a minimum number of gain stages) and otherwise eliminate any other known distortion-causing design characteristics.
The most linear devices known are still triodes. If you are not going to use them, the design field is really limited; you are looking for a zero feedback class A transistor amplifier: Pass Labs, Ayre and Ridley Audio are good places to start.
Transistors themselves are known distortion-inducing devices, so if you allow yourself tubes, the field is considerably larger.
SETs are capable of very low distortion at low powers as they usually don't have hysteresis-induced distortion in their output transformers. Bandwidth and power is problematic and you need higher efficiency speakers to appreciate them. But recent advances in the SET art have made for some impressive gains in transformer design area; this is the edge of the envelope with SETs.
Push-pull amplifiers have power and bandwidth, but suffer loss of low level detail due to hysteresis losses in the output transformer. Two of the more impressive tube designs, especially in light of the fact that they are *not* triode are the CAT amplifier and the Modjesky RM-9. Triode push-pull amplifiers will generally run 300bs or other Directly Heated Triodes (DHT).
OTLs, having no output transformer, avoid the bandwidth and low level distortion issues on that account. This gives them the low level detail of SET combined with the power and bandwidth of push pull, although the bandwidth is usually considerably wider. Choice of speaker is something that has to be paid proper attention as it is with any amplifier.
In general tubes should not be paired with speakers of lower impedance (although this turns out to be true for transistor amplifiers as well, if the best sound possible is your goal). Its a bit of a red herring to fault tubes for not having the 'current' to drive low impedance speakers; the reality is that you want the best tool for the job; butter knives don't make good screwdrivers :)
I apologize for the verbose post, but the subject merits detail as it is at the very heart of what high end audio is (or should be) all about.
