Try this, Matt:
A. If the cover is rectangular or square and runs to the edge of the device
1. Press hard on the inside edge of the battery cover. This should be the edge furthest away from where it slides off the main body.
2. Holding the pressure, slide towards the edge.
3. To refit, study the edge of the battery compartment and the underside of the cover. Match them them and apply pressure, then slide until the lock clicks and the back end of the cover is flush with the side of the device.
B. If the cover is round and fits flush
1. Press hard around the edge until you feel some give under your finger.
2. Hold the pressure and rotate counter-clockwise.
3. To refit, study the edge of the battery compartment and the underside of the cover. Match them them and apply pressure, then rotate clockwise.
C. If neither of the above works
1. Inspect the sides of the device, pressing to see if the plastic is thinner and more flexible at any point. Press hard when you find the spot and insert a small screwdriver to keep the clip open, using the minimum force.
2. On the other side of the device, there will be either a tongue fitting into a slot, in which case the two parts have already parted, or a similar bendy clip as the one you have a screwdriver marking directly opposite. Insert another small screwdriver to press the clip inwards.
3. Refitting is the reversal of this procedure, with the two parts clipping together with minimum force against the weak battery spring.
Good luck.