Given the minimal available output ports (just HDMI and TOSlink optical audio) on the Apple TV, it's no surprise that the low-cost HDMI to DVI cables used in some home media setups aren't supported. Unfortunately, if you're using HDMI converters or cables of the sort, you may be out of luck with the new device.
Your iTunes movie and TV purchases may be available to your Apple TV via Home Sharing, but unless the signal pathway between your ATV and your display is all-digital and HDCP compliant, that content (flagged as 'protected' by the content providers) will not display. While some DVI-equipped TVs and HDMI/DVI cables are HDCP savvy, most are not.
This isn't unique to the new Apple TV among Apple's products, by the way. Try playing a rented/purchased iTunes movie on a DisplayPort-equipped MacBook Pro while a DVI monitor is connected; you'll get bupkis. Even the original Apple TV has HDCP control on the HDMI port, although in that case the analog outputs are available.
Even though Apple's specification is clear on the HDMI all the way requirement, it implies that unprotected content should play fine even if the HDCP checks are unavailable. That's not what several users have discovered, per Ted Landau at Macworld, as they report wavy displays using HDMI–DVI.
If you've been having HDCP-driven agita with your new Apple TV, let us know below.