I'm interpreting the language of your question to mean read every third element, starting with the first one. That's one situation where a good old "c-style loop" comes in handy:

use strict;
use warnings;
use feature qw/say/;

my @array = qw/0 1 2 3 4 5 6 7 8 9 10 11 12/;

for ( my $idx = 0; $idx < @array; $idx += 3 ) {
    say $array[$idx];
}
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perlsyn discusses the various loop types. Your splice method is destructive to the original array, and requires that a new array be set up to accommodate the elements you've selected. The method I demonstrated leaves the original array untouched. If you actually want a new array, you could set one up and push the elements into it within the loop.

Update: You may be asking to read three elements at a time, if I read the language of your question a different way. It's possible to read three elements at a time by just using offsets within the loop, and this would still be non-destructive to your original array.