Being a newbie, please could anyone point me in the right direction with the selection of a cheap accelerometer and any example sketch.
I want to use this to detect any movement, however slight, and trigger an interrupt on my Arduino UNO.
So, if the accelerometer (sensor) moves, I can print "moved" on the Serial Monitor.
I have looked at numerous ICs, but can't seem to find one that has just a digital output when any of the 3 axis change.

The ADXL345 provides interrupts when various different changes in acceleration are detected, but it is not straightforward to set up the device properly. You would be just as well off using the Arduino to analyze the raw accelerometer data. After all, you still have to decide what "moved" means.