Greetings all,
I am currently working with Access myself though I am using a System DSN instead of the DSN-less approach.
Though this is not related to your posting I would suggest using $dbh->{RaiseError} = 1 which will report your errors to you without having to call  || die $handle->errstr all the time for every method call. This may help you figure out where your issue is (db handle vs. statement handle, etc)
One other thing to keep in mind is that the return values from any given query are not what you might be expecting (at least they were not for me). For instance a successful Select query returns me -1 (which in VB is true) even if there was nothing returned, whereas a successful Insert statement returns me 1 (which Im assuming is how many rows were affected); all of this is coming from the method calls on prepared statment handles.
Coming from MySQL mainly this took some getting use to. That being said, use the RaiseError property and post what errors come back, we should be able to help you more after that.

-InjunJoel