Your handy graph paper is 2d. That's certainly how it's been done, and for some that alone is reason to stick with it. But consider what Jaquay himself has used to create maps - various 3d programs. They're complex and often costly.
You do have a free, relatively simple, alternative - google sketch up. There are even video tutorials.
So I set out to do this thing. You can see the results below - click to embiggen.
The origin of these levels was a dwarven mine with a main entrance at the bottom of a cliff on a river. From that entrance, I've got passages to two different wings and a canal the dwarves used to ferry goods in and out - making a shortcut to the far end of the map. There's also a surface entrance, and soon connections to the underdark and a necromancer's catacombs; various levels and sublevels of a megadungeon. This is still in progress and I've yet to add sink holes and other evolutions - but that's easy enough to do.
Melan demonstrated how loops make nonlinear dungeons in his dungeon map analysis. I wanted to make them in 3 dimensions. The dwarves have various vault rooms, with balconies (the level in yellow) accessible from other parts of the dungeon. The forge level (purple), beneath the two shown, has elevators all the way up to the balcony (with a surface entrance) and an elevator down to the mines.
Here's the map as it currently stands with all levels shown.
Here's the water entrance level. The water is in green and you can see the bridge that connects the wings. They'll be a connection between the wings on the mine layer as well.
The balcony level. Some of the entrance level is inaccessible except through the balconies.
The verdict: it's a lot more work than my usual method (see below.) Typically, I grab someone else's map, load it onto my tablet pc and use windows journal to draw on the map. But in that case I'm limited to someone else's layout and without the complexity I want to create.
What a mess. Hard to tell, but there are links on this map to another, and parts are accessible only from the other level - well, you could cross a chasm to get there as well.
What comes to mind then is to combine the two methods. Take a map from Turgenev's maps off his Paratime cartography site (thanks!) and import THAT into sketchup, and then make the 3d slides, chutes and ladders between them. Like this:
There's obviously a lot to sketch up that I'm not using. But for this I don't need textures, google earth, or even objects. I only need this to show the relationship between levels. I've got a mind to take Jaquay's the "Walking Wet" from his early Judges Guild days into sketchup just so that I can understand how the levels fit together!
I definitely won't be using sketchup for every dungeon - I just don't have time or inclination. But if this is really to be a megadungeon the way Gary intended, then the investment could be worth it.