One suggestion I'd offer is to trap the '<Motion>' event as well, to see the window move any time mouse button 1 is down (not just the final release).

You can do that with these minor modifications to your solution:

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1196818

use strict;
use warnings;
use Tk;

# golux:  flag to detect mouse button 1 pressed
my $b_mouse1 = 0;

sub xy { $_[0]->XEvent->x, $_[0]->XEvent->y }

my $mw = MainWindow->new;
$mw->overrideredirect(1);

$mw->Label(
  -text => 'Press 1, Move, then Release in green Canvas to move window
+',
  )->pack(-fill => 'x');

my $c = $mw->Canvas(-width => 500, -height => 400,
  -bg => 'green',
  )->pack;

$c->Tk::bind('<1>' => \&leftdown);
$c->Tk::bind('<Motion>' => \&motion);       # golux:  also detect Moti
+on
$c->Tk::bind('<ButtonRelease-1>' => \&leftup);

$mw->Button(-text => 'Exit', -command => sub { $mw->destroy },
  )->pack(-fill => 'x');

MainLoop;

my ($startx, $starty);

sub leftup
  {
  # golux:  moved the "heavy lifting" into subroutine motion(); the
  # only thing we need do here is clear the mouse button 1 down flag.
  $b_mouse1 = 0;
  }

sub leftdown
  {
  $b_mouse1 = 1;
  ($startx, $starty) = &xy;
  }

sub motion {
  $b_mouse1 or return;
  my $oldposition = $mw->geometry;
  my ($endx, $endy) = &xy;
  my $deltax = $endx - $startx;
  my $deltay = $endy - $starty;
  #  golux:  Not enough to look for \d+, we need to look for [-\d]+, s
+ince
  #  the (X,Y) coordinates can also be negative
  my $newposition = $oldposition =~
    s/\+\K([-\d]+)\+([-\d]+)$/ ($1 + $deltax) . '+' . ($2 + $deltay) /
+er;
  $mw->geometry( $newposition );
  print "$newposition\n";
}
[download]

Instead of moving the window at the end, this moves it any time motion is detected; the only thing needed when mouse button 1 is released is to clear the flag $b_mouse1.

I should elaborate on one thing -- when I tested this (in Windows) I noticed that dragging the window too far up or left caused it to become "stuck". The reason turned out that the one (or both) of the (X, Y) coordinates were negative, which needed to be checked for in your regex to overcome that problem.