FYI, I contacted Jeff Urlwin via CPAN Request Tracker and he suggested to add

$exec->bind_param(1, undef, &SQL_VARCHAR);
[download]

which worked fine.

And then I found out that the "problem" was not related to Perl 5.8 at all. All the time I had DBD::ODBC 0.28 in my 5.6.1. I think I DID try the other versions in 5.6.1, but it seems I did not :-( So now I can confirm that the module behaves the same under 5.6.1 and 5.8 and the difference WAS caused by different version of DBD::ODBC. Sorry to waste everyones' time.

Jenda