My luck ran out on me. Syberia gives only a black screen for me, using Wine-1.1.41. This was the game I expected so much here on GoG and wanted to play out again. This "black screen but music is playing" issue is reported by many. I'm desperately waiting for a solution. I couldn't run the game in Virtualbox, either.
As for Syberia II, there is more hope: the demo runs fine for me under Wine.

I also tried to run GOG's Syberia using latest wine under Linux. I did not succeed (also black screen but music playing...)

After some fiddling around with the retail (2CD) version of the game, I could start it in Wine: AFTER applying a no-cd "patch" (it's very common when using Wine, since Wine doesn't handle most of the copy-protection methods).
GoG's Syberia can also be started this way...if you find the proper "no-cd" thingy.
Is this version here on GoG really, completely DRM free? ..or Mr. Cracker did a better job of making the game Wine compatible than GoG.

By the way, Syberia II from GOG works well under Linux (vanilla wine 1.1.42 + 'winetricks vcrun6' to solve issue with missing msvcp60.dll)
I've been playing it for a few hours, and so far I only encountered sound glitches on the main menu. Appart from that, the game seems to run perfectly.