This answer is a summary of Rabbi Jachter's writeup on this subject.
He provides four (and a half) justifications for why putting pesukim to music is permissible.
The first is that the prohibition was only for Shir Hashirim, because if it is put to music, it is more prone to being misinterpreted as a simple love song. (suggested but not accepted by Igrot Moshe YD 2:142; 1963) The article says this is also implied in Avot D'Rabi Natan (perek 37), but i personally didn't see it.
The second justification is that it is only assur if it's done as a mockery, even if it's not being done as a mitzvah (m'sameach chatan v'kallah). Therefore, most cases should be permissible. Rav Yaakov Emden seems to think along these lines in his commentary on the passage in Sanhedrin.
The third justification (brought unsourced) is that in today's world, when non-Jewish music is so readily available and popular, to davka listen to Jewish music helps enhance your connection to Hashem. Therefore, we can even consider this to be for the purpose of a mitzvah!
The fourth (also brought unsourced) is simple survival as an Orthodox Jewish community. Now that we've moved out of the shtetls, non-Jewish influences are much stronger, and having a strong cultural identity -- in addition to religious practices -- help us preserve our lifestyle.
Also, he writes, פוק חזי -- go see what the common practice is. Because the lenient opinion is so popular, it must be correct (based on Ramban Devarim 17:11).
If you wish to be machmir on yourself and not listen to Jewish (or any) music, that's also permissible, but for me, bring it on!