Our district does allow Dropbox, and I've done mini-lessons with both teachers and students on using it. I think it's a valuable tool for helping students learn to be responsible with their files by backing things up, and it can save teachers from that unpleasant situation of "The files I need for today's class are on the flash drive I left at home / lost / etc." It can also be helpful for collaboration - teachers or students working on the same project can use Dropbox to share their relevant files.
Our district uses a "handout" folder on the school network to share resources. The disadvantage is that you have to be in the school to access the resource.
If your district won't allow access to a dropbox website, consider getting your own domain where you can temporarily upload files. Another option is a mail list where you can upload files (I like Google Groups).
TDSB uses drop off and pickup boxes (among other network features). These features are available to ALL teachers and administrative staff. Additionally, our board has rolled out a collaborative network system called AW, where school communities offer interactive web sites, video/voice conferencing, wikis, online workgroups, share spaces, pickup and drop off assignment folders, custom calendars, unlimited file storage, IM, grade delivery systems and access to school home drives available from school or at home.
Personally, I use the Moodle CMS system for course delivery and AW for the back end system tools (i.e. tying into Trillium student accounts).
If not for any other reason (and there are MANY) explain to your board how dropoff and pickup boxes save the environment (no more photocopying).
I don't see any reason why dropbox should be blocked. My district does not and I use it and a few of my students have accounts for transferring file between home and school for their webpages. I do not let them install the dropbox program on the computers in the lab.
I don't understand why we block things. Isn't it better to educate them how to be responsible? I mean they will always do my space, facebook, youtube, etc. but they need to learn how to use it responsibly and that there is a time an place for it.
I understand districts want to protect themselves but this does more harm than good