ssokolow: It took me a little experimenting to get this working under Wine but, now, it runs perfectly, so I thought I'd share the instructions.

For those who don't need step-by-step instructions, just install wine-staging 1.9.23 and make sure you have CSMT turned on. (Plain Wine gives you a black screen and staging without CSMT scrambles the draw order, giving you flickering or invisible GUI elements.)

Also, using a 64-bit WINEPREFIX seems to cause harmless "crash when quitting" behaviour, so I suggest that PlayOnLinux users choose a 32-bit prefix to avoid the spurious error dialog it'll show.

For those who do need step-by-step instructions:
1. Install PlayOnLinux
2. Select "Manage Wine versions" from the Tools menu
3. In the "Wine versions (x86)" tab, select "1.9.23-staging" and click ">" to download and install it
4. When the process finishes, close the version manager window.
5. click "Install" in the PlayOnLinux toolbar
6. Click "Install a non-listed program" in the bottom-left corner of the install window.
7. Click Next, select "Install a program in a new virtual drive" and click Next again
8. Type a name like "guild_of_dungeoneering" in the field and click Next
9. Check "Use another version of Wine" and click Next
10. Select "1.9.23-staging" and click Next
11. Select "32 bits windows installation" and click Next
12. If the "Wine Gecko Installer" dialog pops up, click "Install" and wait
13. Click "Browse" and pick your Guild of Dungeoneering installer, then Click Next.
14. Click your way through the GOG installer as usual, and just click OK if you get some error popups at the end. Those are harmless.
15. Click Exit in the GOG installer, not Launch
16. Choose "dungeoneering.exe" from the list and click Next.
17. Type a name for your launcher, like "Guild of Dungeoneering", and click Next.
18. Choose "I don't want to make another shortcut" and click Next.
19. Right-click the new "Guild of Dungeoneering" launcher in the PlayOnLinux main window and choose "Configure Wine"
20. Select the "Staging" tab, check the "Enable CSMT for better graphic performance" and then click OK.
21. Your copy of Guild of Dungeoneering should now work perfectly.

Now, if you've got DLC, here's how you install it:

1. Select "Guild of Dungeoneering" in the PlayOnLinux list
2. Click "Configure" in the toolbar
3. In the "Miscellaneous" tab, click "Run a .exe file in this virtual drive"
4. Select your GOG installer for the DLC
5. Click your way through the GOG installer as usual
6. Click Exit in the GOG installer, not Launch
7. Close the configuration window.
8. Your DLC is now installed and playable