Acoustic magnetic generator.

[verpies]

What about this setup (the rod is graphite which is conductive but not magnetic, so it would need to be a similar ferrite rod):

That size and proportion would be much better.  The rod would have to be ferrite not graphite, of course 

One thing that's missing in this arrangement is the massive counterpoise on the other side of the piezo. 
The piezo must brace itself against something either mechanically or inertially.  The mass of the piezo is not enough to serve as a counterpoise (and it's bad for it).

An identical ferrite rod on the other side of the piezo would work as a counterpoise but it would affect the magnetic field a little because it is magnetically permeable, ...yet not an intended part of the magnetic circuit. 
Of course the counterpoise can be made a part of the magnetic circuit, too.   
If the counterpoise is not a part of the magnetic circuit then any hard paramagnetic material is fine ...even a glass rod or aluminum rod, as long as it has the same time-of-flight as the ferrite rod on the other side.

If a different material is used for an inertial counterpoise, then its length should be adjusted according to the speed of sound in it.
For example, aluminum has a high speed of sound, so an aluminum rod counterpoise should be longer than the ferrite rod, in order to make the reflection from its open end come back to the piezo at the same time as the reflection from the open end of the ferrite rod.


                     Speed of       Speed of       Loss factor         Loss factor
                    Longitudinal      Shear         Longitudinal          Shear
Material            waves          waves            waves               waves
---------------------------------------------------------------------------------------------
Diamond      12000m/s
Aluminum       6374m/s        3111m/s     0.00003-0.0001     0.0001
Hard Ferrite   6300m/s 
Steel              5960m/s       3235m/s      0.00002-0.0003
Iron               5957m/s        3224m/s     0.0001-0.0004        0.0002-0.0006
Soft Ferrite    5700m/s
Copper          4759m/s        2325m/s     0.002                      0.002
Brass             4372m/s        2100m/s     0.0002-0.001       <0.001
Gold               3240m/s       1200m/s      0.0003
Lead              2160m/s         700m/s      0.05 - 0.3                0.02

The table above is missing a column listing the slower longitudinal extensional speeds, which are more applicable to long rods. 
For example for a long Aluminum rod the extensional speed is only 5000m/s.  See here and click on "Speed of sound, solid phase".

WARNING: Graphite is porous, soft and has a high acoustic loss factor.  Its spongy structure makes it work better as an absorber than an acoustic reflector.

[verpies]

By "a toroidal core that you can cut with a Dremel tool + diamond wheel ?" you mean
1) cut it in half, or
2) only one cut where the piezo would tightly fit in?

Both is feasible.
In the 1^st case, the piezo would go in one gap and the magnet in the other gap.
In the 2^nd case both the piezo and the magnet (magnets) would have to go in one gap.

Both cases are nicely illustrated here. (the pickup winding can be anywhere on the core. It can also span the entire circumference of the core for some experiments)

[verpies]

In those experiments it's good to have a flat or concave horizontal surface that will hold fine sand in order to cheaply visualize the formation of acoustic standing waves.  Similar to this video but not in 2D nor transversely - only linearly (1D) and longitudinally.

I grinded a shallow groove in a rod to hold the sand when I was doing those experiments in school.

[itsu]

Both is feasible.
In the 1^st case, the piezo would go in one gap and the magnet in the other gap.
In the 2^nd case both the piezo and the magnet (magnets) would have to go in one gap.

Both cases are nicely illustrated here. (the pickup winding can be anywhere on the core. It can also span the entire circumference of the core for some experiments)

What about using a ferrite magnet core like from a big speaker and use the one cut approach to insert the piezo only?
Or is the magnetisation wrong of such a speaker magnet?

Regards Itsu

[verpies]

What about using a ferrite magnet core like from a big speaker and use the one cut approach to insert the piezo only?
Or is the magnetisation wrong of such a speaker magnet?

Unfortunately the magnetization is wrong.
It would have to be magnetized circumferentially, but it is axially