What causes hammering from a new diverter when other faucets are quickly turned on and off?
I'm not sure why you're suddenly having trouble with water hammer, the old plumbing may have been installed in such a way as to mitigate water hammer. Since I have no way of knowing what the old plumbing looked like and/or what you changed, I can't say for sure why you're having trouble.
So I'll simply talk about a couple ways of stopping the hammering.
The old way
We [plumbers] used to try to stop water hammer with 12-18&quot; capped risers, and this method worked great, just long enough for us to be on to the next job.

The idea was that air would be trapped in the capped riser, which would provide a &quot;cushion&quot; for the fast moving water.  This effect, however, is short lived. The air is eventually absorbed into the water, and the riser becomes completely filled with water.
The new way
Modern water hammer arrestors separate the water from the air, which allows them to function properly for much longer. Most devices either use a flexible bladder, or a piston type design to cushion the water and prevent hammering.

These devices are usually less than 6&quot; long, so they can fit in much tighter places. Cost wise they are comparable to the old method, so they are not going to break the bank. They are available with different connector types, so they can be connected to the existing plumbing in a variety of ways.

As with the old style standpipes, water hammer arrestors should be installed in line with the flow of water. Water hammer arrestors should be installed near all fast acting valves, to prevent water hammer.