It's pretty easy on Linux with the compose key. To get "≠", you just hit compose, then "/", then "=". That's actually the same number of keystrokes as "!=" (since "!" requires the shift key).

For whatever reason, the OS documentation lacks a list of allowed compose key sequences. But they are intuitive enough that you can find many of them through experimentation. For example:

Musical sharp ("♯"): compose + "#" + "#".

Interrobang ("‽"): compose + "!" + "?".

Letter "ñ" as in "jalapeño": compose + "n" + "~".

Copyright ("ⓒ"): compose + "(" + c + ")".