I am a Christian, and have attended (and continue to) churches that preach from the Bible, and endeavor to study it and to rightly interpret it.
Also, just in case you aren't familiar with the abbreviations, OT = Old Testament and NT = New Testament :)
Referring to out-of-context passages in the Old Testament (e.g., Leviticus for things about homosexuality) seems pretty common... unfortunately. It is not treated very well, and, as you have noted, pretty much just cherry-picked to "prove" one's point. It's often referred to as proof-texting.
I don't know exactly what you meant, but if you were asking whether that is a good thing - I do not think it is. Israel was a specific nation, with a specific covenant by God; some of the laws appear to coincide with, shall we say, a more universal standard of right and wrong that God has. Others were clearly designed to simply set Israel, as a nation, apart from other nations (e.g., not wearing clothing with mixed fabrics, or many of the dietary restrictions).
In general, I side with non-Christians who criticize Christians for pulling out random OT passages when it suits them and not other ones. Those who criticize Christians for that do, in fact, have a valid point. Christians should not simply pick the parts they like from a law given specifically to Israel and pretend they apply universally to humanity.
Now, that said... I am also of the opinion that there are pretty clear statements in the NT on things like homosexuality (and other sexual sins). And, to be fair, there are parts of the OT that are *not* specific to Israel, even if they were written TO Israel. A lot of stuff in the prophecy books (Isaiah, Jeremiah, Daniel, etc.) has not been fulfilled. A lot of the Psalms, which are just songs, were written by Israelites (primarily David) but often talk about God in generic terms, not Israel-covenant-specific terms... though again, one has to be careful when trying to bring out an application for a New Testament Christian.