You can do this algorithm either on a sheet of paper or in your mind:
In every puzzle there is one line with only one color. If you cross that one out (replace the colors with the blank fields), it becomes apparent that now other lines have only one color, too. Then you cross these out as well and so on...
If you do it on paper you can make a note which color you have crossed out in which line.
If you do it in your mind, you probably can't do this, instead you should remember which lines are gone so you know which line you have to fill first when you do the actual puzzle.
You can go on like this until no colored field is left. There should still be one or more lines you haven't "used"; you can fill them with any color you like.

Now you can start with the actual puzzle: Either you can fill the puzzle according to the notes you made (you have to do it in reverse order, of course) or you know which lines you have to fill first from which point the rest of the way becomes more obvious.

I recommend doing it on paper a few times first and then doing it in your mind, because it gets rather boring, if you do it on paper again and again. Also it's a good memory training to remember which lines you already crossed out.

You can do the crossing-out in the puzzle as well: the outcome should be that all fields go blank (like when you do something wrong). Then you could keep in mind which lines you did last, restart the puzzle and do these lines first.

An example of the algorithm follows (Set 3, Wave 3, Puzzle 25).

I'm going to use this puzzle with the following encoding of the lines:

I have put each step of the algorithm next to it's equivalent in the actual puzzle:
b, black

b, black; 5, brown

b, black; 5, brown; 4, black

b, black; 5, brown; 4, black; c, blue

b, black; 5, brown; 4, black; c, blue; 3, brown

b, black; 5, brown; 4, black; c, blue; 3, brown; d, black

b, black; 5, brown; 4, black; c, blue; 3, brown; d, black; 2, black

b, black; 5, brown; 4, black; c, blue; 3, brown; d, black; 2, black; e, brown

b, black; 5, brown; 4, black; c, blue; 3, brown; d, black; 2, black; e, brown; a, blue

Since 1 is still untouched, we add it to the line, which then reads:
b, black; 5, brown; 4, black; c, blue; 3, brown; d, black; 2, black; e, brown; a, blue; 1

If you put the ribbons in this order:
1, black (any color will do)
a, blue
e, brown
2, black
d, black
3, brown
c, blue
4, black
5, brown
b, black
the puzzle is solved.
(Of course different orders work as well. You could for example swap a, blue and e, brown.)