Spelunky

If you take your time and be extremely patient this achievement won't be too difficult. Do not explore for anything. This only increased your odds of stumbling across stray gold. Watch the video below.

Achievement basics:
To get this achievement you must play from 1-1 through 4-4 without picking up any gold or jewels. The achievement unlocks during the short cutscene after 4-4. If you do obtain any treasure, the achievement is void; losing the treasure by spending it or letting a monkey steal it from you will not help.

This shouldn't be attempted until you've beaten the game a few times so that you're very familiar with the gameplay, items, and levels. There are two factors which make this the highest-ratio achievement in the game:

1. Because you can never buy items, you'll necessarily be less well-equipped if you're attempting a Low Scorer run. You're limited to what you find along the way.
2. It's easy to pick up gold without noticing it. Although gems are large and conspicuous, gold ingots are fairly small and can be hidden behind the decorative bits in the level. Gold nuggets can sometimes be found inside tiles and are nearly impossible to see.

Here are some tips for each area:

Mines
* Explore as much as you can. There are no foreground elements that can obscure gold ingots, and you aren't losing much time if you do accidentally pick up some and have to start over. Whip gold and gems out of your way if you have to.
* Get as many crates as you can. If you have at least four ropes left and can trade one to reach a crate, do it! There may be more ropes in there.
* Go for the Mattock if you get a "snakes" event. Get to the bottom of the pit, whip the rubies all the way to the right, and place a bomb one space to the right of the Mattock. You should be able to pick it up and get out without any gold.
* Sacrifice a damsel to Kali if you get the chance. This is your best shot at getting a helpful item like the Compass, Climbing Gloves, or Cape.
* Try to get the Udjet Eye. There are sometimes valuable items in the walls, and you will also be able to avoid bombing out gems if you make a shortcut.

Jungle
* Watch out for gold ingots on top of trees; the leaves obscure them. Use your whip to flush them out before you walk over the center space. Everywhere else, the ingots are plainly visible.
* Check for crates in pools of water.
* Exploding frogs may scatter gold nuggets from the terrain. If you see evidence of an explosion, be careful!
* If you get the "dead are restless" event, check for the ASH tombstone with a shotgun underneath. And look for vampires; you can take their cape after killing them.

Ice Caves
* Hurry through this area the most quickly. Aliens and mines can destroy terrain and loose tiny gold nuggets. It's not worth exploring.

Temple
* If you've gotten this far, there is more danger from dying than accidentally picking up gold. However, be careful passing behind decorative columns which may obscure gold ingots.

Olmec
* Luckily, Olmec crushes most treasure beneath him. But if the terrain is uneven, some of it may be left. Pay close attention. You may want to use a larger, shallower pit. It depends a lot on which items and how many ropes you have left at this point.

Lastly, if you've had a hard time, don't give up! You might get lucky and find a jetpack in a crate early on.

Just thought I would point out that getting monkeys to pick your pockets will NOT work. Even if you have a score of $0 after beating Olmec you will not get the achievement if you have ever picked up a piece of treasure.

Good solutions above me but I have a method that is able to give you any item available in the shop without angering the shopkepper; ultimately making your runs through the levels a lot easier.

If you play the game with another controller signed in, you can have the second controller pick up an item you want in the shop. As long as you finish the level with the second controller holding that item, when the next level starts, you will now have that item in your possession. If its a jetpack or a cape, the item will be duplicated for both players to have. If its a bag or box of bombs, you will get double the amount you pick up.

Another good thing about playing with two characters is that controller 2 can hold an item or damsel and then you can hold the second player as hes holding that item. This is ideal if you get a dark level because if controller 2 is holding a torch and you are holding him, you can run by the candles and not worry about collecting gold because they wont light.

Edit: This method still works however with the recent title update, you will need to clear your cache and play off line.

Best of luck

***Since the title update in December 2012, the Mattock does not last forever anymore if you take it to the next level. It only lasts 40-60 hits, then breaks (regardless if you took it to the next level or not). I still recommend getting one for this achievement, though.

My best advice is to hope you get a "I hear snakes... I hate snakes!" event in the Mines. In these events, there will be a pit with treasure and snakes in it. At the bottom of the pit will be a Mattock. Use bombs to get it, but be careful not to collect any treasure. The Mattock will last 40-60 hits, then break, so use it wisely.

At the Olmec fight, use your Mattock. Make a gap wide enough for Olmec to fall into, and make sure you have some rope. Be careful though, gems and gold can pop out of the ground. Use the Mattock slowly.

Also, your whip can move treasure out of your way, in case you are being blocked by it.

I do not know if this helps but i earned this one in co-op. Remember your partner must NOT pick up any gold. Though what we did (my brother and I) was simply killing one player at the start of the 1st level (so as to pick up his bombs and ropes). Then i did a decent run but i had SERIOUS and IMPORTANT help by my brother's ghost:
-The ghost can blow away any gold that it is in your path.
-It can blow and stun enemies
-It can trigger the arrow traps
-It can blow and open boxes that you would otherwise risk spending bombs or ropes for them.
-At Olmec's temple (after having collected approximately 30 bombs) i just blew up a 5 blocks wide hole and the ghost helped blowing away any gold that was in the way

First things first, this is NOT EASY. This took me about 75 tries, even though I have a lot of experience with this game. But, here's some tips I've put together to help...

-Stay on the main path. If there's a crate or the damsel is somewhere off the path, don't go for it. You could pick up some gold by accident, and it wastes time, which leads me onto my second tip...

-Beat the levels as fast as possible. If you can beat a level within 20 seconds or less, you will not get a dark level, which can ruin your run easily. If you do run into a dark level, take it extra slow, and make sure to throw your torch down a pit before you venture down there.

-Get the Udjat Eye, only if its relatively safe to get. The Udjat Eye isn't essential, but it'll make you be able to see gems in the walls. This can help you when you are blowing up a wall, so you know exactly where any loose gems are in a wall. It will also help with Olmec, for the reasons I just stated. And speaking of blowing things up...

-Use your bombs carefully. If you decide to plant a bomb, when it explodes gold and gems could go flying everywhere, so get some cover as soon as you plant that bomb. And also, save your bombs and ropes too. You WILL need them for the Olmec fight.

-Don't rob the shops. Now normally, I'd encourage you to rob them, but considering you have to be fast so you don't get dark levels, and the shopkeeper will drop some gold when killed, I can't recommend it. You're not going to need anything he sells, so just rely on crates that you'll come across.

-With Olmec, lure him to the far side of the arena, and find a spot where there's not much treasure in the ground. From there you should place a bomb in the middle of two blocks so it creates a 4-block wide hole, then repeat all the way down. Use your ropes continuously to avoid treasure, and climb to the top of the arena if you need more supplies.

-I know this last one is kind of obvious, but remember, YOU CAN WHIP TREASURE OUT OF YOUR WAY! Seriously, this is the most important tip for this, and without it it would be near impossible.

Thanks for reading, and I hope this guide will help you grab this stupid achievement.

Happy Spelunking!