I also own a Linn Classik hooked up to a pair of Tyler Reference Monitors. I haven't done much with it in terms of replacing cables yet (still using the factory speaker cables as well.)
However, I always throw away those pre-molded cheap 18 gauge IEC cords that ship with everything from computers to audio equipment. They are just plain junk. I always replace them with at least standard 14/3 gauge powercords.
Not sure how much much improvement it made as I never even listened to the unit with the original powercord. It's also probably not the best upgrade you can do for a powercord, but it's a start and cheap.
However, I have had lots of luck and experience with the Granite Audio #555 and #560 powercords (which I use throughout all of my main system.) They're a lot powercord for the money.
I'm curious what you've found that works for speaker cable. The Linn Classik has those bizzare speaker connectors that almost nothing fits except for those z-connectors. Not one of the best choices in my opinion given its a really very nice integrated unit.
Thanks.