As another example that uses dividers or a compass (or marking a radius on a sheet of paper) rather than a straightedge, see the nomogram on the last page (marked page 101) of the 3 pages I scanned into this Word document: http://www.myreckonings.com/Temp/HypotenuseNomogram.doc
This excerpt is from Das Entwerfen von graphischen Rechentafeln (Nomographie)
by P. Werkmeister, Berlin, 1923.
The x and y functions for each scale of the nomogram are shown to the left of that nomogram on page 101. To use it you place one leg of the divider/compass (or the corner of a sheet of paper) on the y-scale value, then set the second leg (or mark the paper) at the x-scale value, and then swing that second leg around to hit the z-scale value (or move the sheet of paper to match the marked interval).
The problem with all the solutions here so far is that they all involve powers of x and y, so the scale distances increase or decrease drastically with x or y. But this is in contrast to simply using regular graph paper, where you have x and y plotted linearly and you just find their intersection and measure with dividers/compass/paper from the origin to that intersection and lay it down on the x or y axis to measure the length of the diagonal. This linearity provides greater accuracy over a much greater range of x, y and z than any of the nomograms we've mentioned here. Is this a case where nomograms are inferior to other graphical methods, or are we missing some other solution?