OK, I now have a working SleepyHead-1.0.0-beta2 on my new Linux system. It's a pretty generic distribution (Slackware 14.2), so the instructions should work for anybody.
I'm using 1.0.0-beta2 on my Windows machine, but thought I'd try the beta0 from jedimark's page. Got a license (free) from GitLab.com and downloaded it. Converted the .deb file to tgz with deb2tgz and unpacked it with the Slackware package tool. It "installed" but failed at run time due to a lack of the Qt5 library bits. OK, time to start from scratch. ("Use the Source, Luke")
Research:
Run glxgears. That works, so I have openGL.
Run qmake --version. That returned Qt 4.8.7. Oops.
Read the jedimark pages. (These went unavailable after Christmas while Mark is rebuilding his page, but the basic bit is that Qt5.5.1 is optimal, and anything higher than 5.5 will break SleepyHead.)
Initial setup:
1) The machine wants to use the / partition, but mine ran into space problems (cough*LibreOffice*cough), so I made a large /usr/local partition for builds. If you don't have space problems, skip the /usr/local and carry on.
Getting the pieces:
Getting the Qt5.5.1 is the most involved, but is fairly easy once you've navigated the route.
Go to Qt.io; hit the download button. You now have to answer a few questions to get the bits. You want the "in-house deployment" option. Once it's happy, you get to the downloads. Go the "other files" and the archives. You'll want the qt5.5.1 section, where you'll find the open source file. This is everything, so you shouldn't need Qtcreator. Download the 600Mb+ .run file and move it to the right spot in your machine (/usr/local for me.
Make the run file executable with chmod 744 and run. You will now have Qt5.5 in an opt directory (/usr/local/opt for me). Find the main qmake location. Mine was /usr/local/opt/Qt5.5.2/5.5/gcc_64/bin.
I wanted to keep Qt4.8.7 for other applications, so I finagled the qmake command.
cd /usr/bin
qmake is a simlink to the distribution Qt4 make, so I renamed it
mv qmake qmake-4.8
Then, ln -s /usr/local/opt/Qt5.5.2/5.5/gcc_64/bin/qmake qmake-5.5
Finally, ln -s qmake-5.5 qmake
qmake --version points to the right one. (Once I'm done, qmake will be redirected to qmake-4.8)
Now, get Sleepyhead source. Log in to Gitlab, then in a terminal, run
git clone https://gitlab.com/sleepyhead/sleepyhead-code.git
Move the code to an appropriate directory, and unpack it.
Check the README for buidling in the shadow directory. I found that libudev-dev was not necessary; the libudev pieces in my distribution were sufficient. You should end up with SleepyHead in the build_sleepyhead directory. Run it, and you are good.
End note: I'm moving from a Windows box to a Linux one, and to keep my data, I went to the Windows SleepyHeadData folder, went down to the "Peter" folder for my data and profile, and copied it to the other machine.