Hmmm, as Corion has pointed out, you'll get much better response by showing what you have tried, and in what way it fails.
in reply to Regex to find repeated patterns
That being said, does this code do what you are looking for?
my $str = "*L*Name1*L*Name2*L**D*Place1*D*Place2*D**L*Name3*L*";
my @names =$str =~ m/\*L\*(\w+)/g;
$str =~ s/\*L\*$names//;
print "Modified string = $str\n";
Name1 Name2 Name3
Modified string = *L*Name2*L**D*Place1*D*Place2*D**L*Name3*L*