What is the quickest excavation method without TNT?
Don't forget that better (or worse!) pickaxes affect your digging speed. Using Diamond pickaxes may speed things up in the long run, (gold is even faster, I believe?) and wood pickaxes will give you intermittent breaks in the monotony as you retreat to get a new pickaxe.
In addition to Raven Dreamer's answer concerning tools, you can "power mine" sand and gravel. The below applies to both sand and gravel.
Sand normally exists as a block. When a sand block's support is removed, it turns into a sand entity and falls down until it hits another object, at which point it places itself as a block in the nearest on-grid position, or turns into a dropped sand item if that position is occupied (such as by a torch).
I use this method to harvest sand for glass intensive projects by placing torches under the sandstone/stone and then breaking the sandstone/stone. Viola, instant 4-6 sand blocks. It works very quickly, but you have to dig down to the sandstone/stone layer underneath the sand. Obviously this creates large pits where the sand was.
Source:Sand/Gravel Falling
When I dug a pit to bedrock, this is the technique I used after I ran out of TNT.
Overall plan
Dig a vertical shaft down to the bottom level (use safe techniques to avoid dropping into a cave) and place ladders to climb up.
Dig horizontally to make a 2-block-high space covering all of the horizontal area you're planning to dig out. This tells you when to stop digging later, and also checks for deep caves filled with lava. (Note that lava lakes, which can occur at any height, are still a hazard.
From the top, dig down around the edges (vertically) of the area (techniques covered below). This checks for caves intersecting the region you are digging.
Having made these cuts, you now have a better idea of what hazards the remainder contains that you might encounter while digging out.
If you are not entertained by being the proud owner of a slightly floating landmass, proceed with digging out the middle.
Vertical digging techniques
Caution: These techniques are optimized for simplicity, not safety. You should get a brief glimpse of any caves but you might not react fast enough.
While standing on top stand just on the edge of one block while facing another block, and aim so that your line of sight intersects both. You want to be positioned such that both blocks are supporting you, so that when you click to mine, you remove both blocks before you actually fall.
Then hold down the mouse button. You will repeatedly mine two blocks and fall one block. When you reach bottom, you will fall 3 blocks into your previously cleared space (taking no fall damage since it's less than 4).
If you have an L-shaped set of 3 columns, you can position yourself to do the same all-at-once digging trick, but there's no way to do it with a 2×2 (since your aim can't pass through all four).
