What does it mean that physics no longer uses mechanical models to describe phenomena? I've just started reading Sommerfeld's Lecture on Mechanics, with no background in physics (only in math). Can you explain to me what the author means with the bold sentence?

Mechanics is the backbone of mathematical physics. Though it is true
  that we no longer require physics to explain all phenomena in terms of
  mechanical models, as was common during the last century, we are
  nevertheless convinced that the principles of mechanics, such as those
  of momentum, energy, and least action, are of the greatest importance
  in all branches of physics.

 A: "Mechanical models" probably refers to the ideas that Maxwell had later in his life, in which he visualized space as being populated by tiny gearwheels which were enmeshed in each other. He thought that electromagnetic effects were propagated through space by the linked rotations of those gearwheels. 
A: I believe it's because as we've dived deeper into physics, we realized that the behavior of matter and energy can no longer be modeled by mechanisms.  Instead, mathematics has to stand in for the old tangible, touchable, mechanical models.  In the physics of the past, matter and energy were conceptualized as single, individual particles that had intrinsic properties of their own, and were so small (in some cases, sizeless) that when acting in concert, give rise to the material world we experience.  However, modern developments such as quantum physics and string theory show us that...


*

*The reality of physics is not in absolute truths, but in probability.

*The entities that physics is attempting to explain and model, even when probability is overwhelming, aren't exclusively point-particles, as a mechanical model would have you believe.  These entities sometimes behave like waves, which is much more difficult to model with a mechanism (and much easier to model in mathematics).

A: Just an example. There were times when physicists tried to explain electromagnetic forces using mechanics. Something like "there is some media which fills the space, in presence of electric charges this media can be stretched or compressed hence we have some forces". It was a common belief that true explanation of various phenomena must be mechanical, like this one.
But then they realized that this visual explanation actually explains nothing. Because the nature of elastic forces in materials is electromagnetic forces between atoms. You can't explain electromagnetic forces via electromagnetic forces.
Sorry, I do not remember the source of this information.
A: I think the answers above are excellent, but I'd like to point out a related issue.
During the early 20th century we developed two entirely new types of physics, GR and QM. Its difficult to imagine two theories that were more different from each other. GR is essentially classical mechanics in a non-Euclidean geometry, QM is, well, still being debated.
So for much of the middle of the 20th century you saw the QM people trying to quantize GR, and the GR people trying to "geometrize" QM. So we had the idea of gravitons, the quanta of gravity, as well as twistors, the geometry of particles. Neither worked, and we're still largely where we started off in spite of much effort (and strings, supergravity, etc.).
The parallel is important. Classical mechanics remains spectacularly successful. So when you start thinking about something like "heat", the first thing you do is try to re-use your existing models, and presto, you get a formula for heat transfer that actually works... mostly. But as time went on we saw that some things simply didn't work that way no matter how hard we tried, like radioactivity, and eventually we stopped trying to apply mechanics to absolutely every problem.
And thus the quote. We no longer try to apply some version of Newton's original axioms to every problem because we know they aren't universal.
A: It's likely the author is referring to concepts including and similar to Rutherford's planetary model of the atom.  In this model you have electrons orbiting the nucleus of the atom like planets orbit the sun.
Modern physics dispenses with this model because it simply doesn't really exist.  These orbits are replaced by a probabilistic wave function.  However we still talk about 'spin' despite the fact that we know that particles are not little balls and they don't physically spin on an axis.
