Bulky Chubby Heavy Foam Flies
There are some huge stonefly and hopper-like patterns that can be hard to get moving through the air with light lines like 3.5 level line or less. They also have a tendency to twist your tippet as you cast.
Oddly, I've found that some big flies aren't a problem to cast- mainly large say size 10 or 12 Stimulators and Elk Hairs. I think the elk hair is more dense and cuts through the air better than giant rectangles of foam and polypropylene yarn wings.
If you use a light size 3 level line or less the fly acts like an anchor and just doesn't move with your cast. If you manage to get it in the air it swings more like a medieval ball and chain than an elegant dry fly.
Some would advise not to fish a big hunk of foam, fish a kebari (wet fly). But we're fishing to have fun. And big foam flies are fun to watch a fish crush on the surface.
Use a furled or tapered line - they can transmit force from the thicker butt section all the way down to the tippet and bulky fly turning it over. But expect the fly and line to land hard, possibly spray when you strike or recast, and sag so your drag free drifts will be shorter.
Use a shorter line than the length of the rod - this give you the ball and chain situation. But, now you can launch the big bug.
Water haul - let the fly drift down stream of you until the line comes tight and then catapult the fly forward to it's target, you might even get one false cast in with this technique. Accuracy is reduced a bit.
Use a PVC line - there are several on the market that appear to be the running line sections of dry fly floating western lines. They are heavy and so they will lie on the water just like with a western fly rod set up. This is the method advocated by Patagonia and Blue Ribbon Flies. It also explains the super stiff rods that Western Tenkara rod companies are building like TFO and The Tenkara Rod Co. This is my least favorite solution.
Make a line using Cortland 20lb test monofilament braid. I got this idea from a DIY project from Tenkara Talk (check it out here). I use the 20lb line because the 30lb was way too heavy and sagged. The 20 lb can be used as your entire line, but I prefer to use it as the butt section of my hybrid line. Basically 1/2 to 1/3 braided mono - then a section of level line then tippet. It also makes avery good wind line. You will get a little bit of line sag, but much less than the other alternatives.
Shad caught this nice brown using a Kiyose 14 foot rod with the cortland hydrid line to cast a heavy streamer and egg dropper and hook into this nice brown trout.