Don't hang on to everything. Yes, some things will need to be filed away for future reference. Those art projects from three terms ago and the piles of dry erase markers that haven't worked since the Clinton administration
aren't among them. Once a semester, go through everything (and we mean EVERYTHING) that is taking up extra space in the classroom.
File away neatly anything that will be of use to you, and throw out everything else. For things that hold sentimental value like Christmas cards and letters from students, start a scrapbook and separate sections based on school year. This way you have one nice compact space to keep valuable items while still keeping the classroom neat and tidy.
Don't let your students hang on to everything either! How often do you make them go through their school desks and cubbies? If it's not at least once a week, you can be sure that there are piles of old test papers and forgotten homework balled up somewhere in the back corner. Each week before you all leave for the weekend, have students clean out their personal space. Added bonus: Wait until this day to hand out any papers that need to be signed by parents. That way, they can't use the excuse of "It got lost in my desk".
Use the cubby system. Yes, the tried and true cubby method is still the best way of keeping a classroom neat and tidy. One variation: use removable tubs instead of traditional cubby holes. That way students can search for lost items more efficiently, overflow is kept to a minimum, and they add a bit of pizzazz to the classroom if you choose ones with bright colors!
Try and get the rest of the school in on the action. Perhaps a monthly "cleanest classroom" contest. Offer the students a prize...like a free homework pass for each child or maybe an extra recess at the end of the day. With so many eager students in on it, the whole school will be spotless in no time.