Out of curiosity, what kinds of things are you not allowed to bring to the Academy as a cadet? Things like movies, music, phones, etc.

From the Academy's FAQ's:
6. What items are cadets allowed to have in their rooms?
You’ll not be permitted to bring your personal possessions with you when you enter the Academy (except for a few items listed in the cadet appointee instruction booklet). All basic necessities, such as uniforms, bedding and linens, will be furnished when you enter. During the academic year you’ll be permitted to have additional items when authorized by the Cadet Wing Commander. You’ll be issued a personal computer while at the Academy. You’ll be permitted to have a radio or stereo equipment in your room beginning the spring semester of your Fourth-Class year; you must wait until your First-Class year to have a television in your room. You may, at certain times, watch TV in the squadron recreation room. You must wait until your Second-Class year to have most electrical appliances, but a Third-Class cadet may have a coffee pot.

That is pretty accurate. Also, most sophomores can get permission to have a mini-fridge in their room.

Freshmen before recognition are not allowed to have media (DVDs, video games, music recordings, etc).

Weapons are prohibited in the dorms.

Other than that, the restrictions are mostly based on what you can fit in your room.