The Pillar Clip method can be done without Zora Armor or the Air Replenish Trick by sinking to the bottom of Lake Hylia while it is under twilight (requires Early Master Sword). The clip through the pillar and reaching the loading zone before you run out of breath is extremely difficult. The following video is the old school setupless method which does not call Midna prior to the clip. This method is one of the hardest ways to perform pillar clip and has since been obsoleted by modern techniques, covered in the next paragraph.
Modern setupless tutorial:
There is now a set-up for pillar clip without Zora Armor created by dragonbane0 which is very consistent.
If you play on a CRT you have to adjust the visual cues.
There is a strat that skips the first visual cue for the Pillar Clip setup by Rodner which saves a few seconds. After skipping the cutscene hold down-right and mash a to swim up.
As soon as Link starts swimming, hold up and keep mashing A so Link will swim towards the bridge. After the Midna text, swim towards the texture shown in the video.
From there continue with the normal setup.
By using a cutscene to stop the air meter, you can do Pillar Clip in high-water Lake Hylia using a Cucco.
Swim over to the GCN left side of the entrance to the temple between the large blue pillar and the green pillar. Position yourself so that the yellow arrow on your mini-map is pointing GCN north-west. You can use the Iron Boots to get into position and then take them off to make it a bit easier. Hold the Target button, and swim straight upwards. Make sure to mash A to increase the swim speed. You should clip through the blue pillar and be able to swim into the loading zone without having to blow up the rock.
A rupee dive can only be done with a blue rupee or higher. The quickest way to spawn one is to dig up a patch of dirt close to the dive spot, but one will not always appear.
Once a blue rupee appears, use the gale boomerang to bring it to the ledge near the entrance of Lakebed Temple. Take out the sword, hold Target (sometimes helps get a better camera angle), and press B a split second before you get the rupee. The timing is hard. If you swing your sword too soon, you'll get the rupee in midair and the animation won't happen to where you can dive out of bounds. However, this is easier than standing on the very edge, not swinging your sword, and hoping the dive works. From there, swim your way into the tunnel.
Swim to the middle of Lake Hylia where the floating wooden platforms are located. Get on one of the platforms and use the Gale Boomerang to bring a wooden box to you. Get on the very edge of the platform, and prompt Midna. You drop the box, which will then push you off the wooden platform, causing you to cutscene dive out of bounds. From here, swim your way into the loading zone.
A dive from "low" Lake Hylia is also possible using a pot from Fyer's hut. This isn't faster but is easier and also allows you to get into Lakebed earlier.
After knocking the Boss Bug into the water, you can call Midna on the same frame the Boss Bug disappears to Midna Dive. From there you can swim up into the Lakebed loading zone
Using Ooccoo at the edge of the planks in the water, you can dive down under the floor collision