Since I have switched to Linux, replacing AHK with equivalent Linux apps has been a sort of Holy Grail for me. AHK is a Swiss-army-knife-type utility for Windows and there is just no replacement for Linux. Sure, Autokey is always trotted out as a replacement but it is a) not nearly as powerful as AHK and b) a lot less reliable (not to say buggy). Nevertheless, there are some things Autokey does well most of the time, not least global hotkeys and system-wide menus both of which can execute user-written Python scripts. There's a significant learning curve but then again there's also a significant learning curve for AHK

Autokey is in the repos although the latest 0.95.10 (test repo) has a couple severe bugs that are (mostly) fixed in git head.
One thing Autokey can't do is using mouse buttons as triggers (it can replay them though). Enter a second utility: xbindkeys (in the repos) can be used to start programs, scripts etc for any keypress or keypress combination... including mouse buttons (also all non-standard mouse buttons so long as X recognises them). In the long run I will try to replace Autokey with xbindkeys.
Number three is devilspie2 (also in the repos). This small utility can be used to position and size all sorts of windows on the current desktop or move them to other workspaces. It can also trigger arbitrary other actions whenever a window is created. Again, there's a learning curve.
Last but not least there's something I discovered just a few weeks ago: espanso. This is a superb utility to create hotstrings, next to hotkeys one of the seminal strengths of AHK. (A hotkey is just a systemwide key combination that simply triggers an action whereas a hotstring is a (normally short) typed string that is dynamically expanded to another string or triggers some other action.) Espanso does this fast and, most importantly, very reliably. This is a real gem AFAIC since I had a collection of about 500 hotstrings under Windows for all sorts of things... email addies, pathnames, login names, some throwaway passwords, account nos... you name it. Espanso handles all that and much more gracefully. It is also not that hard to use, at least for the basic hotstring capability. Absolutely recommended.
So... these four utilities between them can almost completely replace AHK under Linux , at least for my use cases. It is not exactly pretty, it requires some perseverance, but it works. YMMV.