Consider a small piece of solder epoxied to the needle of
a phono pickup for high audio frequency accelerations.

Consider a speaker with a lead weight glued in center and
lots of holes punched for lower frequencies.

Make an (RF) tuned circuit tuned by a small coil with a sliding
slug. Suspend slug on small brass coil. Read oscillation frequency
off into comparator, divider, and into pic.

For DC:

Now, a 3D version:

Get a hollow cylinder and fill with alcohol (water will get bacterial
growth),
put a pressure sensor on each side of each axis (one at each pole and 4 on
equator).
Pressure on sensor is proportional to the acceleration in that direction.

eliminate sensors in uninteresting directions. If you only want 1 direction,
just use a
length of plastic tubing, if you know the direction of acceleration.

For a "one bit" maximum value test -
suspend a ball bearing in a cage of rubber bands. If the ball falls out,
you've exceeded the limit.

all this depends on:

a) how many bits?  What data rate? What frequency response? How rugged is
rugged? What g range?

For many applications, a singe pass/fail bit is enough, and for many
applications a maximum value is enough.

almost any physical quantity can be measured by converting it to either
resistance or light intensity (and thence to
resistance), or by changing the physical dimensions of a tuned circuit, or
by constructing a battery.

max value, pass fail equipment can be quite crude. They sell shipment
monitors commercially that you stick on
the shipment - they have a glass vial with a membrane, if the membrane fails
the chemicals on opposite sides mix
and they turn color. These cost pennies each.