You could use focusing rails used in macro photography.
This should be ok for X and Y axis but for Z you should add "something"...
I do like it (and what I was thinking of), but the one's I'm seeing of this type aren't cheap (4 way using lead screws; $650
example).
Bit pricey for a DIY project IMHO.
There are at least some inexpensive dial indicator holders (
example #1,
example #2; different armature types).
To DIY the latter type, flexible coolant tubing used for CNC might be suitable (
Loc-Line or
Jeton). Doesn't offer an ultra precise Z adjustment (assuming the use of a cheap 4 way focus rail for X & Y <
example>), but should work IMHO. Just need a bit of stone, metal, ... for a base (attached via an appropriate NPT flange) and rig up a clamp for the probe.
Should also work if just using the flexible tubing to set the position (lose the fine precision adjustment entirely). Certainly becomes affordable this way at any rate, and if you don't like it, it can be repurposed as a helping hand for other tasks.