Since you've already got a good Perl 5 answer, here's a Perl 6 answer (that works with present day Rakudo):

my @arr = (0,0,1, 3,2,0, 1,1,1);
my @output;
for @arr -> $a, $b, $c {
    @output.push: $a + $b + $c;
}

say @output.perl;
# output: [1, 5, 3]
[download]

Or a bit more idiomatic:

my @arr = (0,0,1, 3,2,0, 1,1,1);
my @output = gather {
    for @arr -> $a, $b, $c {
        take $a + $b + $c;
    }
}
[download]

(This one would even be lazy if Rakudo did laziness yet).