and it's faster for not having to generate an extra array:

use warnings;
use strict;
use Benchmark qw(cmpthese);

my @arr = (0,3,4,5,8,10,12,15, 1..999999);

cmpthese (-2,
        {
        mapgrep => sub {map {$arr[$_]} grep {$_ & 1} 1..$#arr},
        slice => sub {@arr[ grep {$_ & 1} 1..$#arr ];},
        grep => sub {my $ind = 0; grep {$ind++ % 2} @arr},
        push => sub {my @var; push(@var, $arr[$_*2+1]) for 0..int(@arr
+/2)-1},
        map => sub {map { $arr[$_*2+1] } 0..int(@arr/2)-1},
        }
    );
[download]

Prints:

# my @arr = (0,3,4,5,8,10,12,15, 1..999999);

          Rate mapgrep   slice    push     map    grep
mapgrep 2.02/s      --    -29%    -35%    -38%    -40%
slice   2.84/s     40%      --     -9%    -13%    -16%
push    3.11/s     54%      9%      --     -5%     -8%
map     3.27/s     62%     15%      5%      --     -3%
grep    3.37/s     66%     18%      8%      3%      --

# my @arr = (0,3,4,5,8,10,12,15);

            Rate    push mapgrep   slice     map    grep
push    160309/s      --    -30%    -39%    -50%    -54%
mapgrep 229948/s     43%      --    -12%    -28%    -34%
slice   261705/s     63%     14%      --    -18%    -25%
map     319334/s     99%     39%     22%      --     -9%
grep    350258/s    118%     52%     34%     10%      --
[download]

The performance holds pretty much regardless of size.

Update: more cases added

I was curious how using map and multiplication to generate the slice, without any conditionals, would compare. ("mult" in this example.) Interestingly, map is killing grep on my machine, which differs from what GrandFather got.

#/usr/bin/perl

use warnings;
use strict;
use Benchmark qw(cmpthese);

my @arr = (0,3,4,5,8,10,12,15, 1..999999);

cmpthese (-2,
          {
        mapgrep => sub {map {$arr[$_]} grep {$_ & 1} 1..$#arr},
        slice => sub {@arr[ grep {$_ & 1} 1..$#arr ];},
        mult => sub {@arr[map {$_*2+1} 0..int(@arr/2)-1];},
        grep => sub {my $ind = 0; grep {$ind++ % 2} @arr},
        push => sub {my @var; push(@var, $arr[$_*2+1]) for 0..int(@arr
+/2)-1},
        map => sub {map { $arr[$_*2+1] } 0..int(@arr/2)-1},
        }
    );
[download]

This is a multiprocessor x86-64 machine, for what it's worth.

~/perl$ cat /proc/version
Linux version 2.6.26.8-57.fc8 (mockbuild@x86-4.fedora.phx.redhat.com) 
+(gcc version 4.1.2 20070925 (Red Hat 4.1.2-33)) #1 SMP Thu Dec 18 18:
+59:49 EST 2008
~/perl$ perl -v

This is perl, v5.8.8 built for x86_64-linux-thread-multi

~/perl$ perl ./everyother.pl
          Rate mapgrep   slice    push    mult    grep     map
mapgrep 5.24/s      --    -22%    -28%    -33%    -34%    -43%
slice   6.73/s     28%      --     -8%    -14%    -15%    -27%
push    7.28/s     39%      8%      --     -7%     -8%    -21%
mult    7.80/s     49%     16%      7%      --     -1%    -15%
grep    7.92/s     51%     18%      9%      1%      --    -14%
map     9.22/s     76%     37%     27%     18%     16%      --
[download]

Update: As an added note, ($_<<1)+1 is slightly faster still, but doesn't catch up to the map and grep solutions.

my @arr = (0,1,2,3,4,5,6,8,10);
             #     #     #

# Giving
$VAR = [1,4,8];
[download]

my @arr = (0,1,2,3,4,5,6,8,10);
my $ind = 0;
my @result = grep {1 == $ind++ % 3} @arr;
[download]