Pyth - 3 bytes

Very simple, will add expansion later, on mobile.

s.T

Test Suite

CJam, 4 bytes

qN/z

Try it online!

We can also write an unnamed function for 4 bytes, which expects a list of strings on top of the stack:

{zs}

Try it online!

Perl 6, 34 32 bytes

{[~] flat roundrobin |$_».comb}

{roundrobin(|$_».comb).flat.join}

A lambda that takes an array of strings as argument, and returns a string.

(Try it online)

Python 2, 101 89 86 69 bytes

I'm hoping I can get this into a lambda somehow, shortening it down by making it recursive. It isn't ideal because you would hope transposing is shorter, unfortunately it isn't (from what I have managed to come up with so far).

f=lambda s:' '*any(s)and''.join(x[:1]for x in s)+f([x[1:]for x in s])

Old solutions:

w=input();o=''
while any(w):
 for i in range(len(w)):o+=w[i][:1];w[i]=w[i][1:]
print o

lambda s:''.join(''.join([c,''][c<' ']for c in x)for x in map(None,*[list(y)for y in s]))

w=input();o=''
while any(x>=' 'for x in w):
 for i in range(len(w)):o+=w[i][:1];w[i]=w[i][1:]
print o

^{thanks to mathmandan for making me feel dumb ;) saved me a bunch of bytes! (on an old solution)}

C, 114 84 bytes

-20 bytes for not calculating the length.

i,b;f(char**s){b=1;while(b){i=-1;b=0;while(s[++i]>0)if(*s[i])putchar(*s[i]++),++b;}}

Accepts array of char pointers and requires last item to be a null-pointer (see usage).

Ungolfed and usage:

i,b;f(char**s){
 b=1;
 while(b){
  i=-1;
  b=0;
  while(s[++i]>0)
   if(*s[i])
    putchar(*s[i]++),++b;
 }
}


int main(){
 char*a[]={ 
//  "POLLS","EPEES"
//  "LYES","APRONS"
 "ABCDE","a c","123 567"
 ,0};
 f(a);
 puts("");
}

Haskell, 33 bytes

import Data.List
concat.transpose

Try it on Ideone. Usage:

Prelude Data.List> concat.transpose$["ABCDE","a c","123 567"]
"Aa1B 2Cc3D E567"

Without using a build-in: (38 34 bytes)

f[]=[]
f x=[h|h:_<-x]++f[t|_:t<-x]

Try it on Ideone. 4 bytes off thanks to Zgarb! Usage:

Prelude> f["ABCDE","a c","123 567"]
"Aa1B 2Cc3D E567"

Retina, 13 bytes

Byte count assumes ISO 8859-1 encoding.

O$#`.
$.%`
¶

Try it online!

Explanation

O$#`.
$.%`

This is based on the standard transposition technique in Retina. We sort (O) all non-linefeed characters (.), by ($#) the number of characters in front of them on the same line ($.%`), i.e. their horizontal position.

The second stage then simply removes linefeeds from the input.

PHP, 68 bytes

for(;$f=!$k=$f;$i++)for(;""<$v=$argv[++$k];$f&=""==$c)echo$c=$v[$i];

Loops over command line arguments. Run with -r.

After the inner loop, $f is 1 when all strings are finished, 0 else (bitwise & casts ""==$c to int).
Next iteration of the outer loop: copy $f to $k (saves one byte from $k=0) and toggle $f:
When all strings are done, $f is now false and the loop gets broken.

JavaScript (ES6), 52 46 bytes

f=([[c,...s],...a])=>s+a?c+f(s+s?[...a,s]:a):c

Takes input as an array of strings and outputs as a single string.

Test snippet

f=([[c,...s],...a])=>s+a?c+f(s+s?[...a,s]:a):c

g=a=>console.log("Input:",JSON.stringify(a),"Output:",JSON.stringify(f(a)))

g(["SIMPLE"])
g(["POLLS","EPEES"])
g(["LYES","APRONS"])
g(["ABCDE","a c","123 567"])
g(["\"\\n$?*",",(.)\" "]) // Backslash and quote are escaped, but in/output are correct

Actually, 7 6 bytes

Golfing suggestions welcome! Try it online!

Edit: -1 byte thanks to Teal pelican.

a Z♂ΣΣ

Ungolfing

          Implicit input each string.
a         Invert the stack so that the strings are in the correct order.
<space>   Get the number of items on the stack, len(stack).
Z         Zip all len(stack) strings into one, transposing them.
♂Σ        sum() every transposed list of chars into strings.
Σ         sum() again to join the strings together.

JavaScript (ES6), 46 bytes

f=([[c,...s],...a])=>c?c+f([...a,s]):a+a&&f(a)

<textarea oninput=o.textContent=f(this.value.split`\n`)></textarea><div id=o>

PHP, 77 bytes

Golfed

function($a){for($d=1;$d!=$s;$i++){$d=$s;foreach($a as$v)$s.=$v[$i];}echo$s;}

Anonymous function that takes an array of strings.

I'm sure this could be golfed more, but it's early. On each iteration, we grab the i-th letter from each given string and append it to our final string, one at a time. PHP just throws warnings if we access bits of strings that don't exist, so that's fine. We only stop when no changes have been made after looping through all the strings once.

I feel like the usage of $d can be golfed more, but it's early. :P

J, 13 bytes

({~/:)&;#\&.>

Try it online!

Based on the inspiration for this question.

Another way to do it takes 27 bytes but operates using transpose. Most of the bytes are to handle the automatically added zeroes from padding.

[:u:0<:@-.~[:,@|:(1+3&u:)&>

Explanation

({~/:)&;#\&.>  Input: list of boxed strings S
          &.>  For each boxed string x in S
        #\       Get the length of each prefix from shortest to longest
                 This forms the range [1, 2, ..., len(x)]
                 Rebox it
(    )         Operate on S and the prefix lengths
      &;         Raze both
   /:            Grade up the raze of the prefix lengths
 {~              Index into the raze of S using the grades
               Return

Bash + GNU utilities, 55

 eval paste `sed "s/.*/<(fold -1<<<'&')/g"`|tr -d \\n\\t

I/O via STDIN (line-separated) and STDOUT.

The sed formats each line to a bash process substitution. These are then evaled into paste to do the actual interleaving. tr then removes unnecessary newlines and tabs.

Ideone.

C, 75 71 bytes

Only limitation is the output length. Currently it's 99, but can be easily stretched to 999 (+1 byte).

i;main(a,b)char**b;{a--;for(;i<99;i++)*b[i%a+1]&&putchar(*b[i%a+1]++);}

Ungolfed:

i;
main( a, b )
char **b;
{
    a--;
    for( ; i < 99; i++ )
        *b[i % a + 1] && putchar( *b[i % a + 1]++ );
}

Python, 112 bytes

i=len(x)if len(x)>len(y) else len(y) h=0 a="" while h<i: if h<len(x) a+=x[h] if h<len(y): a += y[h] h+=1 print a

Java, 19+155=174

import java.util.*;String f(Queue<String> q){String r="";while(!q.isEmpty()){String s=q.poll();r+=s.isEmpty()?"":s.charAt(0);if(s.length()>1)q.add(s.substring(1));}return r;}

Ungolfed:

import java.util.*;

  String f(Queue<String> q) {
    String r = "";
    while (!q.isEmpty()) {
      String s = q.poll();
      r += s.isEmpty() ? "" : s.charAt(0);
      if (s.length() > 1) {
        q.add(s.substring(1));
      }
    }
    return r;
  }