The Catalyst 7.11 Linux Driver is able to detect these graphics cards as being based upon the RV670 GPU, but you can likely expect proper support in the Catalyst 7.12 driver release later this month.
We've also shared with you before that adding Radeon HD 3850/3870 support to the open-source RadeonHD driver shouldn't be much of a problem, according to what the developers had to say. We've tried it by adding in their respective PCI IDs, but it wasn't as easy as that. With the latest git code right now, it will then error out on the TMDSBVoltageControl, because the 3850/3870 aren't listed in the RV6xxTMDSBMacro array. Long story short, we're waiting on hearing from the Novell developers when the proper open-source support may arrive.
Expect a full review of both the ASUS EAH3850 TOP and EAH3870 TOP later this month. There are already active discussions about the ATI Radeon HD 3800 series going on in the Phoronix Forums.