For some time it was thought that this achievement was broken and reports indicate it was at one point. It does work now though and it is one that some may be confused on.
The main thing to understand here is what the game considers a bonus. Equipping artifacts isn't a bonus. Using items you find in gold chests or get via trading with Hindu that are needed to complete a level are not a bonus. What is a bonus is when you use items like the Horn of Plenty and the level doesn't require you to use it to complete the objectives. So if your objectives say something like use the magic shield 3 times, as long as you only use three magic shields, you're fine. I found it a good idea to just avoid opening the gold chests or trading for them with the Hindu unless the level objectives required it (and even then, I saved those for last). The reason for this is that selecting things near the top of the screen, like what to build on a lot, can accidentally trigger a bonus if you aren't careful.
You also have to do this on every level, that includes the alternates. It doesn't have to be done the first time. As long as you complete each level without using bonuses at some point the achievement will unlock. Nothing in the game keeps tabs on this, so if you miss it you may have to replay all levels. This is definitely one that is best to get as you go and to replay a level if you did use a bonus. Also not all levels can be completed in the required time for a gold coin without a bonus, so there will be a few you may need to redo after getting the gold coin to do with no bonus. The achievement will unlock once you return to the map after completing the last level needed to finish without bonuses.