What are some of the half dozen, or so, most popular jewish, or hebrew, songs?

Judaism is a religion while Hebrew is a language.  Jewish songs that
are most well known have to do with holidays and traditions and are in
a variety of languages.  Popular Hebrew songs could be whatever is on
the radio right now in Israel.  You may get a better answer if you
specify exactly what you are looking for.

well,
I can answer you question,since I am an Israeli jew.
but your question,is not specific enough since I'm not sure if you are
talking about one or more of the above :
1. modern songs
2. ancient songs
3. religious songs
4. zionist songs

I think you're not talking about modern songs,cuz the modern songs are
not that unique,So I can give you some others,and if you want some
specific species of music then just clerify :)

1. Hava Nagila.
2. David Melech Israel.
3. Am Israel Chai.
4. Besiman tov (Siman tov u mazel tov).
5. Yerushalaim Shel Zahav.
6. Hadegel Sheli Kachol Velavan
7. Ken Ani Od Chai (by Uzi Chitman)

there're really alot of them,but this 7 should gives you an indication :)