A developer known as tx00100xt made a source port of the Serious Engine to Linux that works fine.
The first approach would by using it:

https://github.com/tx00100xt/SeriousSamClassic

It also support many mods and allows to play Serious Sam Alpha.

Once you compiled / installed the source port; create the following directories:

/usr/share/serioussam/ - for TFE
/usr/share/serioussamse/ - for TSE

or

~/.local/share/Serious-Engine/serioussam/ - for TFE
~/.local/share/Serious-Engine/serioussamse/ - for TSE

Unpack the gog .exe installer files with innoextract and copy in the directories the *.gro files, the Help directory, and only for the First Encounter the Levels directory as well.

See here:
https://tx00100xt.itch.io/serious-sam-classic


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Otherwise if you prefer, both encounters work well in Wine. You just might need to set the resolution properly, just like in MS Windows.

To setup everthing start the game once and open the configuration file InstallDir/Scripts/PersistentSymbols.ini and fix the following lines:

gap_iDepthBits=(INDEX)24;
hud_bShowMessages=(INDEX)0;
sam_iScreenSizeI=(INDEX)XXXX;
sam_iScreenSizeJ=(INDEX)YYYY;
plr_fFOV=(FLOAT)ZZZZ;

The first two values are fixed and are needed to workaround a game bug that allows you to see through objects and the second one disable Netrisca notifications in the center of the screen.
The other three lines are for the resolution, XXXX is the horizontal value, YYYY for the vertical and ZZZZ if the field of view (90 is the best for 4:3, 100 is the best option for 16:10, and 110 for 16:9).

In the case of problems related to multi CPUs start the game with:
$ schedtool -a 0x1 -e wine SeriousSam.exe

There is an unsupported patch for the Second Encounter, you might want to install it too as the new features are nice; it breaks internet play though. The installer does not work normally with the gog version, but as often what you cannot obtain normally can be done sideways.

Here are the instructions:

- Download the game[0]
- Download the patch[1]
- Download the no-cd patch for 1.50[2]
- Install the game normally
- Open the patch .exe file with 7zip (other archivers might work too) and unpack it somewhere
- Copy the content in the SSSE installation directory
- Unpack the crack, copy the new .exe in the bin directory
- Execute the game once and leave
- Setup the resolution in the .ini file

[0] http://www.gog.com/game/serious_sam_the_second_encounter
[1] https://files.seriouszone.com/ -> classic-tse -> official -> patches -> secondencounterpatch150publicbeta.exe
[2] http://m0001.gamecopyworld.com/games/pc_serious_sam_tse.shtml

Have you tried the native Linux installers?
http://files.seriouszone.com/catdisplay.php?catid=78

I have them downloaded, I just haven't gotten around to trying them out.

Sorry, I did not. I had enough of the Linux version of Serious Sam 2 that worked much worse than the original in Wine; beside the description is somewhat discouraging.
But if there is interest I will give it a try.

Seriously game companies, if you cannot be cool as Id (it is indeed hard) at least support Wine officially. Dang.

Thank you so much for this post! I just found it. You're awesome!

You are welcome. I made similar posts for most gog games I played, nice that someone appreciate.

Really helpful, thank you so much! The game works great!

I use the Linux native versions using the original .run files from Loki

liflg.org/?catid=6&gameid=71

The games run smooth.

While running Serious Sam 1 in Wine the resolution would often change dynamically upon re-opening the game from time to time.

Updated the text to support the native linux client from tx00100xt.

Oh, and there's a Vulkan source port, too. https://github.com/tx00100xt/SeriousSamClassic-VK Here we are.