The game doesn't have any native widescreen support.
This is what you have to do:
1. Get a hex editor. I googled and found a free XVI32 which is doing its job perfectly.
2. Use the hex editor to open game's config file, silkcfg.dat. You can find this file in game's program directory (typically c:\program files\silkroad\). You need to run this game at least once to have this file created.
3. Open silkcfg.dat with XVI32
4. At byte locations 8 and 9 there's a little-endian encoding of your horizontal resolution. At byte locations 12 and 13 there's a little-endian encoding of your vertical resolution.
5. Change the resolution to your native resolution. In my case, this meant changing 40 06 to 80 07 (780 hex = 1920 dec). Use windows calculator in advanced mode to quickly convert numbers.
6. Save the file.
If you broke the file or otherwise you're in doubt, start Silkroad's welcome screen, choose "option" and pick a resolution again. The config file will be re-created. There's no risk. Don't forget to reduce shadow quality from the game later on, or it will be very slow.
The outcome is, game runs at your native resolution.
I was unable to run the game at anything but 1920x1200. When given other numbers (1680x1050, 1920x1080, 0x0) the result was quite random and often resulted in 1920x1200, which actually made me write incorrect things here [this is an edit].
To be precise, this is because silkcfg.dat is not a config file. Config file is setting\SROptionSet.dat. When you modify silkcfg.dat, game notices the change and then re-creates SROptionSet. If you do something wrong, SROptionSet will not be modified.
Also, if you make a backup of silkcfg.dat and copy it back later, it will be considered "old" and ignored. Backup SROptionSet.dat instead.
Only native resolution seems to run.