my $key = join '', @$_;

Note that join-ing with the empty string means that, e.g.,  [ 'hello', 'sun and fun', 'day' ] cannot be distinguished from the arguably different subarray  [ 'hello', 'sun and funday' ] (among other permutations):

c:\@Work\Perl\monks\Anonymous Monk>perl -wMstrict -le
"use Data::Dump;
 ;;
 my @MyArrayOfArray =
 (
     [ 'hello', 'sun and fun', 'day' ],
     [  2,      'okay',        'may' ],
     [ 'hello', 'sun and funday'     ],
     [  2,      'okay',        'may' ],
 );
 ;;
 my (%hash, @AoA2);
 ;;
 for (@MyArrayOfArray)
 {
     my $key = join '', @$_;
     push @AoA2, $_ unless exists $hash{ $key };
     ++$hash{ $key };
 }
 ;;
 dd \@AoA2;
"
[["hello", "sun and fun", "day"], [2, "okay", "may"]]
[download]

A join string that is guaranteed not to appear in any text will avoid this. Here I use  $; (see perlvar), the default value of which just happens to work in this particular case:

c:\@Work\Perl\monks\Anonymous Monk>perl -wMstrict -le
"use Data::Dump;
 ;;
 my @MyArrayOfArray =
 (
     [ 'hello', 'sun and fun', 'day' ],
     [  2,      'okay',        'may' ],
     [ 'hello', 'sun and funday'     ],
     [  2,      'okay',        'may' ],
 );
 ;;
 my (%hash, @AoA2);
 ;;
 for (@MyArrayOfArray)
 {
     my $key = join $;, @$_;
     push @AoA2, $_ unless exists $hash{ $key };
     ++$hash{ $key };
 }
 ;;
 dd \@AoA2;
"
[
  ["hello", "sun and fun", "day"],
  [2, "okay", "may"],
  ["hello", "sun and funday"],
]
[download]

Update: Slightly more concisely:

c:\@Work\Perl\monks\Anonymous Monk>perl -wMstrict -le
"use Data::Dump;
 ;;
 my @MyArrayOfArray = (
   [ 'hello', 'sun and fun', 'day' ],
   [  2,      'okay',        'may' ],
   [ 'hello', 'sun and funday'     ],
   [  2,      'okay',        'may' ],
   );
 ;;
 my @AoA2 = do {
   my %seen;
   grep ! $seen{ join $;, @$_ }++, @MyArrayOfArray;
   };
 ;;
 dd \@AoA2;
"
[
  ["hello", "sun and fun", "day"],
  [2, "okay", "may"],
  ["hello", "sun and funday"],
]
[download]

(Update: Posted this update before I saw Dallaylaen's post with essentially the same idea.)