It's been years since I have last touched 68k Embedded.. So I'm a bit rusty.. But..
http://cache.freescale.com/files/32bit/doc/ref_manual/MC68340UM.pdf brings back memories.
What I'd would start is to check datasheets of the most obvous chips and backtrace/sort (with Ohm meter continuity beep) out the voltages needed for the board and to check power supply (Voltages + ripple/noise) in case the Electrolytes/tantals have gone bad. Obviously there is a lot of +5V parts but most likely there will be some other voltages for analog stuff.
For debugging, it was quite common trick at the time to add some extra I/O just to get 'visible' indicators out of the system. We used to have either Serial port and/or typical 2*16 char epson-controller LCD for system debug purposes. Those help a lot, when you are testing the system. So I'd check if there's any obvious connector/header that might have 68340 UART pins. Also finding Max232 (or derivate) on board would be lucky indeed :-)
Mc340 seems to have JTAG and BGD in it, so if you can find tools for those, they might give indication if the CPU itself is running. Of course simple checking Addr/data lines could tell that much. Those programmable logics may also have Jtag, but I doubt they woudl have been left unprotected.