Several things.

1. $2 and $3 are not populated by simply using /g
2. You're not testing for matches
3. Your __DATA__ section reflects a "log file" which has precisely 3 words per line. Note what happens to line 3 of my __DATA__.

#!/usr/bin/perl
use strict;
use warnings;
# 778252

while(<DATA>) {
       if ( $_ =~ /(\w+\s*)(\w+\s*)(\w+\s*)/ ) {
           print "$1 $2 $3";
    }
}
__DATA__
word1 word2 word3
word4 word5 word6
7word 12345 8word *@- 9word
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Output:
word1 word2 word3
word4 word5 word6
7word 12345 8word