If you are on a slab,remove the base board and notch the slab with a grinder and come out below the cabinet stile to the left of the DW.
You can use copper or nylon and a bead of caulking on top to hold it in position.

Is the house on a crawl space? If so, it may be easier to go under rather than across the room. While some may scoff, SharkBite (and other, similar fittings) make it easy to install a T and valve in an existing waterline. I really hate soldering over my head in a confined space....

I should have clarified. This is in a condo building. I won't be ripping up the new floor for this. I'd also prefer to minimize damage to the drywall.

Ahhh...this could be a challenge, because you don't actually own the walls. The condo association does.... You own the space between the paint on one wall and the paint on the opposite. Same with floor and ceiling.... Read the condo declaration carefully to see what your rights are and what modifications are legal w/o prior approval from the board.

Run it on the wall, behind the dishwasher and across...on the baseboard. Put it low enough behind the dishwasher so the back doesn't crush it.

Protect the line on the baseboard, either by making a groove for it (I would replace the baseboard, actually), or run it on top of the baseboard, then install another piece of trim moulding above that to protect it. I would not "hide" it.

Something like that.

This post was edited by homebound on Tue, Feb 11, 14 at 18:09

Tap in to your cold supply -- hopefully near the dishwasher -- with a tee and a shutoff valve for your refrigerator line. Remove the baseboard molding. If there is a gap at the bottom of the sheetrock that can accommodate a 1/4" copper line, slide it in place. If there is no gap, cut out a small trench in the sheetrock for the line. Connect one end of the line to the tee and the other end to the refrigerator. Reinstall the baseboard, being careful to not nail through the new line. Touchup the paint.

This post was edited by kudzu9 on Wed, Feb 12, 14 at 19:14

Dishwashers have cold supply-lines? None that I've ever owned have.

Gwarstong-
Thanks, I knew that didn't sound quite right when I typed it at 3:30 in the morning. Anyway, I went back and edited it.

"Tap in to your cold supply -- hopefully near the dishwasher -- with a tee and a shutoff valve for your refrigerator line. Remove the baseboard molding. If there is a gap at the bottom of the sheetrock that can accommodate a 1/4" copper line, slide it in place. If there is no gap, cut out a small trench in the sheetrock for the line. Connect one end of the line to the tee and the other end to the refrigerator. Reinstall the baseboard, being careful to not nail through the new line. Touchup the paint."

I've torn the place apart, so the baseboard is gone now. This sounds like the best way to do it given my constraints. Thanks to everyone for their input.