sub LoadNewValues{
%MyValues=();
%MyValues=("1",  "hallo",
            "2", "ok",
            "3",  "fine");
}
[download]

The basic problem with this  LoadNewValues() function from the OPed code is that it assigns a new set of referents to the global  %MyValues hash, but it does not change the referents originally assigned as  -textvariable references in the  $mw->Entry(-textvariable => ...) statements. (I hope I've made myself clear on this point; references are tricky.)

Using something like your original approach, this works as I expect you expected it would:

c:\@Work\Perl\monks>perl -le
"use warnings;
 use strict;
 ;;
 use Tk;
 ;;
 my $mw = tkinit;
 $mw->geometry('200x200');
 ;;
 our %MyValues = qw(1 term1  2 term2  3 term3);
 ;;
 for my $DBcolumn (1..3) {
   $mw->Entry(
     -textvariable => \$MyValues{$DBcolumn},
     -justify      => 'left',
     -width        => 25,
     )->pack(-side => 'top', -anchor => 'w');
   }
 ;;
 my $button1 = $mw->Button(
   -text    => 'Update Values',
   -command => \&LoadNewValues,
   )->pack();
 ;;
 my $button2 = $mw->Button(
   -text    => 'Print Out Content',
   -command => \&PrintOutContent,
   )->pack();
 ;;
 MainLoop();
 ;;
 sub PrintOutContent {
   print qq{$MyValues{1} $MyValues{2} $MyValues{3}};
   }
 ;;
 my $inc_them;
 sub LoadNewValues {
   if ($inc_them++) {
     $MyValues{$_}++ for qw(1 2 3);
     }
   else {
     @MyValues{ qw(1 2 3) } = qw(hallo ok fine);
     }
   }
"
term1 term2 term3
hallo ok fine
hallp ol finf
hallq om fing
[download]

Update: The code using  configure() that you posted here works fine, but does not address the basic conceptual problem with handling references that lurks in the OPed code.