Is this a new build as a permanent dwelling?
So it's a log cabin, with probably 4" solid wood log walls. How does that comply with current building regs regarding insulation? I've come across these as outbuildings, where building regs don't apply, and I've come across them where internally they are battened out, insulated and plasterboarded.
There are a few of these around here, mostly built in the 1970's, but they actually planned ahead when they were built, and (almost certainly in the factory) they drilled cable drops into the logs to hide cabling. Not an option to retro fit and no scope for modifications, but that's how it should have been done.
I do quite a few wooden "eco houses" but they all have standard plasterboard internal walls with a service void behind the plasterboard.
Your only options do appear to be surface mount. If they want to keep the wooden look, what about some wooden dado rail with a channel routed out of the back?
Dare I say for light switches, hide cables behind the door archtrave (cough, safe zones, cough)
Is it a suspended floor? how about rather than hiding cables, make a feature of surface mounted wiring e.g galvanised conduit coming up from the floor to each socket?