ä, ö and ï should display correctly on most, if not all PC systems. They're included in the ASCII standard character set, see ascii table.
However, Apple must have different numbers for each character. To test it, I wrote out a little text file with the "funny" chars, and transfered it to my Mac, and opened it there. See the image attached for the comparison.
So I suggest you rely in the html entities.
I'm sure Linux will support those too.
Garth Farley