The closest and easiest access point for water for our fridge line is to the kitchen sink, but if we run the tubing through the cabinets, it will have to go behind the stove somehow (something tells me that isn't a good idea or against code, right?)
There are no bathrooms or other water sources near the fridge, that I can think of, and the ceiling in the basement is drywall so we can't run it under the floor.
Our former neighbors had their hardwood floor ruined by a leak from the water line being run under their stove. The heat wasn't the problem....something squished it and caused a slow leak.
I wonder if there is some kind of pipe that you could run the tubing through, so that it would be less likely to cause an issue.
Where is the fridge going to be in relation to a utility room? Our fridge is basically on top of the utility room and the water line is run up through a small hole in the floor from a branch in the line that goes to a powder room. The shut off valve for it is in the utility room too.
Cut a hole in the basement drywall and run the line from the basement. Having a tiny bendable line in an open area is never a good idea plus the plastic fittings (or tubing depending on what you install) will crack from the heat of the oven. Refrigerator water lines are a big source of water leaks so would you really want it running through multiple cabinets that could get damaged when it starts to leak? And yes it's a "when it starts to leak" not "if it starts to leak." The tiny 1/4" lines are not meant to last long. Creating an access panel in your basement ceiling under the fridge would be a good idea for easy repairs in the future.