For what it's worth (and as far as I'm aware), H1B visas are now capped (i.e. full) until October 1st, 2012 - applications are accepted from April 1st - so there may not be much sense in companies including the keyword "H1B" as requested, unless they're looking to fill a roll later in the year..
Except H1B is used as a generic term to indicate willingness to sponsor visa applications, including types other than H1B visas (such a TN1 for Mexican and Canadians), so please, keep including it.
Sorry, not trying to be snarky but as someone working on a visa myself, if you need a visa and you work in tech then it pays to spend the ~$200 or so initial consultation fee to speak to an immigration lawyer who can brief you on your options.
Even if your future-employer will handle the immigration work (eg on an H1b), it still pays to retain your own immigration council because their lawyer is only to going to look out for their interests.
