How can I make cool, 3D dungeons? Maybe with clay? Dwarvenforge is waaaaaay to expensive

Clay you can bake in a typical home oven could work. I'm a fan of Sculpey brand clay, and they have a few different firmness options available.

I'm not positive, but I think get a firm clay and some tools or house-hold implements (i use random things like an old knife, a small metal screwdriver, different shaped screws, toothpicks and an old Bic pen) to carve patterns might be the best - if time consuming - route to go.

You could potentially scale-up the operation by using your sculpted clay pieces to form molds for plaster casting. Kind of like doing your own Dwarvenforge thing, which I presume would be less expensive than buying from Dwarvenforge, but would take a much larger chunk of free time.

Or, I've heard there are 3D paper dungeon materials, but I'm otherwise completely unfamiliar with them.

There's quite a few good 'papercraft' companies. I can recommend these:

Fat Dragon

World Works

I haven't tried the following, but they also look good, especially the free downloads!

Papercraft Dungeon 

If you have the time and inclination, you can make some stunning pieces.

Another options is molding them yourselves (as suggested by Aaron). 

Hirst Arts makes the molds: http://www.hirstarts.com/molds/molds.html

And I ran into this company while trying to find Hirst's name. They provide "blueprints' for creating cool D&D-oriented stuff with the Hirst molds: http://mythicaldungeons.com/index.php?route=product/category&path=69

If this Kickstarter has already been mentioned, sorry for the duplication.  The Battlefield Crate has me tempted.

I recommend Tabletop Plus. Good quality pieces that are less than half what Dwarvenforge costs. I think a lot of the stuff is from hirst arts molds and I know they are compatible. You do have to paint your own pieces though. 

http://tabletopplus.com/

if you're a little crafty and wanna safe on money - check out DMGInfo on Youtube - it's mainly 2.5D, but super cheap and rather easy

I've printed a few papercraft pieces. My only problem is my fat hands and trying to glue tiny little tabs together on curved structure pieces or trying to glue on roofs from the inside of a building. They look kinda nice though if the roofs aren't too wonky ;)

I made mine out of cardboard, though now I am also buying some MDF ones.

But cardboard is super cheap and looks good. If I was home I would post pictures of my creations.

I also buy a lot of MDF ruins to paint. I have so many now.

If you search for MDF fantasy or MDF ruins or "Frostgrave" on ebay you will find so many of them that are pretty nice and pretty cheap.

I really like Black Magic Craft's tutorials. I believe he's a contractor so he knows all these really cool "hacks" for cheap materials from hardware stores. -- https://www.youtube.com/channel/UC2Rlv-ug-mtnXuMwlpcqFgg

I used DMG myself. I like that the dungeon tiles are modular and therefore reusable for different dungeons and scenarios. If I knew how to attach photos, I'd add mine. I made them all pretty quickly including some furniture and it made me super proud to have made something. Also it was indeed super cheap. I get excited whenever I see double corrugated cardboard now.