It has occurred to me that there are very few resources available for people who want to take BLD past the Old Pochmann method. I have (for the most part) switched to 3-cycles for corners, and I feel that although I am no expert on the method, I understand the process fairly well. This guide will only address corners, but the same concepts can be applied to edges.
The problem most people have is that the idea of having 378 prepared 3-cycles is terrifying. If you are one of these people, I want you to try as hard as you can to forget that thought. Good. Now you can start learning.
The basis of all 3-style BLD is BH. Most people who have thought about taking their BLD to the next level have gone here:
and given up immediately.
Those who are slightly more persistent have reached this tutorial:
and gotten so confused about cyclic shifts and columns cases that they have given up. Let me tell you something. I almost completely use 3-cycles, and I have no idea what commutators are classified as cyclic shifts.
So, many people have gotten very lost in the land of BH and given up very quickly. I was almost one of these people, but I came across this post by Mike Hughey:
You should read the whole thing, but the part that got my attention was:
I found this to be completely true. Think about learning 3-style like you're learning F2L, and you'll get it very quickly....about 3 weeks ago I decided to go ahead and learn the whole thing. I was very surprised to discover it only took me about 3 weeks! It’s really not that hard. In fact, for an experienced cuber, I suspect it will take you about the same amount of effort as it originally took you to learn Fridrich F2L. Really! So I’m saying that I think it takes about as much effort for an experienced cuber to learn BH corners as it takes for a beginning cuber to learn Fridrich F2L!!!
Anyway, onto how I learned it.
The first thing you need to learn, if you haven't learned it already is how to do pure commutators. Brian's tutorial (linked above) is perfect for this. make sure you understand how to do a pure commutator backwards and forwards. This is the only new concept you will need to learn to shift to 3-cycles.
Do you have a handle on pure commutators? Good.
Now I want you to watch byu's video on orthogonals (skip A9s for now). These cases are solved using a setup move, a pure commutator and undoing the setup move. Easy, right? You should now be able to solve any 3-cycle on the cube using a few setup moves and a pure commutator. Practice this a lot.
At this point you can solve any 3-cycle. It might not be close to optimal, but you can solve it, and it's probably less moves than a Y-perm.
When you feel ready, I want you to chose a sticker on your cube. I use ULB as my buffer, and I chose RFD, which turned out to be very easy, so I recommend that if you use ULB as a buffer choose RFD, and if you use UBR as a buffer, choose LFD. Now come up with a commutator for every cycle involving the sticker you have chosen and your buffer. DO NOT LOOK AT A LIST OF ALGORITHMS. The whole point is that you will remember these algorithms because you came up with them yourself. You can let these commutators be sub-optimal, but try to find ones you like to execute. You can even start looking for A9s at this point, which is just when one of the setup moves cancels. Try to see similar commutators on the same interchange layer. If you're using ULB as a buffer and RFD as your chosen sticker for example, your cycles to FUL, DFL and BDL all use the same interchange layer and thus the commutators are basically the same. It is easy to find groups of three like this.
Now you have 18 commutators that you came up with and understand. I want you to learn them backwards and forwards. Just focus on those algorithms. The goal is to get to the point where any time you get your chosen sticker in a solve, you can use a commutator. Therefore you need to have these at your fingertips. Practice them in attacks. One method that helped me was having a list of the commutators, and every time I forgot one during a solve I would put an X next to it in the list. Then, I would practice the commutator forwards and backwards 5X times, so the first time I got it wrong I would practice it five times, the second time I got it wrong I would practice it 10 times, and so on.
Now you are at the point where every time you get your chosen sticker in a solve, you use a commutator. Congratulations. It is easy from here. All you have to do is choose another sticker on that piece and do the same thing. If you chose RFD for your first one, I recommend choosing FDR for the next one since all the commutators from RFD can be mirrored for use with FDR. Learn them the same way. Good. Now do the same thing with the third sticker on that piece. Great!
You might be thinking that you are nowhere near getting rid of your Y-perms, but you would be wrong. I found that once I knew these 54 commutators (108 including inverses) I could solve any cycle on the cube. All I had to do was use a smart setup move so that one of the pieces would end up in RFD, DRF or FDR and I could solve the cycle.
From there, of course, I did a lot of cleanup (I have a lot more left to do), and I obviously use a lot of cycles that don't involve those three stickers, but that is how I worked my way up to 3-cycles for corners. Edges are next!
I hope my story helped, and good luck!