One option that I've used on a couple of sites is to use SyncBack, JungleDisk, and an Amazon S3 account. (I would post links to these products and services, but SE prohibits us new users from posting more than 1 link. Boo.)
For those not familiar with these tools, SyncBack is an awesome and very feature-rich program for syncing files between different folders. JungleDisk let's you create a virtual drive that, behind the scenes, reads and writes data to Amazon S3's storage service, which is a cloud-based storage solution that's pretty affordable (presuming you're not pushing up gigabytes of data into the cloud every day).
We have SyncBack run on a schedule, like once a night or once a week, and it copies the new/changed files from the local hard drive to the JungleDisk virtual drive, and removes any deleted files from the virtual drive.
What do we store in the cloud? Well, we have SQL Server do a nightly backup and have it configured to keep those backups for two weeks. These backups are pushed up to the cloud every night. We also push up our source code repositories every night. A complete backup is done of static files, assets, utilities, and other such files on a weekly basis.
What's nice about this approach is that all the files are in the cloud, meaning we can access them from anywhere. No need to RDP into the dedicated server if you've got JungleDisk on your desktop - you can see the files in the cloud as if they were on a drive on your local computer. Another benefit is cost. For one of my clients the web hosting company that hosts the site will sell an offsite storage backup service to its dedicated server clients, but the costs was several hundred dollars per month. This approach had some up front costs - the SyncBack and JungleDisk licenses - as well as storage and bandwidth costs from Amazon S3, but, if I recall correctly, these costs were less than what one month of the hosting company's backup plan would have cost. And this way we don't have to take the hosting company's word that our site is getting backed up - we can see our files in the cloud at any time from anywhere in the world.
(I was introduced to this approach by my colleague Michael Campbell.)