The cleanest way is to NOT do any explicit error handling. It may be better to handle your exceptions in one place, using "eval" and Exception::Class::DBI. So when you connect, you can pass this:

...
RaiseError  => 1,
HandleError => Exception::Class::DBI->handler,
...
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Now, if any error occurs, an exception will be thrown, and you can catch that via:

eval {
  # if I fail, I will let you know
  $dbh->do("evil query"); 
};

my $ex = undef;

if ( $ex = Exception::Class::DBI->caught() ) {
   # log error in $ex->errstr
   # show users a general error screen
}
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In general, and totally IMHO, checking for return of every db statement is evil ;)