Flatten the Array!

K, 3 bytes

,//

This is a fairly common idiom. "Join over converge".

try it here with oK.

How it works:

Join (,) fuses together atoms or lists to produce a list. Over (/) takes a verb (in this case join) and applies it between each element of a list, left to right. Thus, the compound ,/ will flatten all the top level elements of the list. The symbol / actually has different meanings depending on the valence (number of arguments) of the verb with which it is compounded. When we provide ,/ as the verb, the final / acts as "converge"- it repeatedly applies ,/ to the input until it stops changing. Some other languages call a feature like this a "fixed point combinator". By repeatedly fusing bottom level lists, you will eventually arrive at a single flat list, and none of the operations will perturb the order of elements. This seems to solve the problem.

JavaScript (ES6), 35 bytes

Inspired by @user81655's answer:

f=a=>a.map?[].concat(...a.map(f)):a

Mathematica, 16 14 bytes

{##&@@#&//@#}&

An unnamed function which takes and returns a list, e.g.:

{##&@@#&//@#}& @ {{{20}, {"Hi"}, "Hi", 20}}
(* {20, "Hi", "Hi", 20} *)

Explanation

Syntactic sugar party!

To understand how this works, note that every expression in Mathematica is either an atom (e.g. numbers, strings, symbols) or a compound expression of the form f[a, b, c, ...], where f, a, b, c are themselves arbitrary expressions. Here, f is called the head of the expression. Everything else on top of that is just syntactic sugar. E.g. {a, b, c} is just List[a, b, c].

We start with //@ which maps a functions over all levels of a list. For instance:

f //@ {{{20}, {"Hi"}, "Hi", 20}}
(* f[{f[{f[{f[20]}], f[{f["Hi"]}], f["Hi"], f[20]}]}] *)

Note that this maps f over atoms as well as compound expressions. What we're now looking for is a way to get rid of the list heads and keep everything else.

The Apply function is normally used to feed the elements of a list as separate arguments to a function, but its actual definition is more general and simply replaces the head of an expression. E.g. Apply[g, f[a, b]] gives g[a, b].

Now there's a special "head" called Sequence that simply vanishes. E.g. {a, Sequence[b, c], d} just evaluates to {a, b, c, d}. The idea for flattening the list is to replace the heads of all inner lists with Sequence so that they get splatted into their surrounding list. So what we want is to Apply the head Sequence to the lists. Conveniently if we Apply something to an atom, it just leaves the atom unchanged, so we don't have to distinguish between types of expressions at all.

Finally, there's one small issue: f is also applied to the outermost level, so that it also removes the outermost List, which we don't want. The shortest way to counter that is simply to wrap the result in a list again, such that the surrounding Sequence can safely vanish.

Note that there's neither Apply nor Sequence in the code. @@ is an operator form of Apply and ##& is a standard golfing trick to shorten the long built-in name Sequence. So ungolfing everything a bit, we get something like:

flatten[list_] := { MapAll[Apply[Sequence], list] }

For more details on how and why the ##& works, see the section on "Sequences of arguments" in my answer for the Mathematica tips.