This achievement is a little misleading. There are only three assult rifles in the game that you can use: The Bar, Bren and MP44. I originally thought that an assult rifle was the same thing as machine gun and I couldn't figure out why I wasn't getting the achievement when using a MP40 (which are everywhere) and my trusty (but crappy) Thompson.
This is how I got the achievement: I did the Island level (level 2) on Easy. You start out with a BAR so right away you are looking good. Conserve ammo! I didn't find any MP44's on the level and obviously there are no Brens so if you run out of BAR ammo, you are hurting at the end part (getting up that hill with no bullets is pretty tough, even on Easy). I was able to really conserve ammo by meleeing every chance I got. Clearing the houses and barns? Melee! Gerry in the open? Melee! All in all it's a pretty easy achievement to get once you get the gist (or is it jist?) of it.
** As a few people have pointed out using granades will not affect getting this achievement so feel free to let pineapples and potatoe mashers fly **