I happen to work in a small village, and I noticed while walking to work that if I were to go along the side of one of the churches here and through the back, I could get to the local supermarket. I wouldn't be saving a huge amount of time by doing so (maybe a couple of minutes at best), but I was wondering if there was a ruling about walking through any part of church property to go somewhere else. The only benefit I would get would be shortening my route to the supermarket, and I wouldn't be using the church at all otherwise.
Within FOUR AMOT is Asur (In Gemara Avodah Zarah 17 it says that one isn't even allowed to go near the entrance of a house of Avodah Zarah. So explained Tosfot on 17. So explained Rambam on his peirush (Perek 1, Daf 11 Amud 2). And so wrote the Tur (Tur, Yoreh Deah 149). And so wrote Maran (Shulhan Aruch, Yoreh Deah 157). Also the Ritv"a, in his Hidushim (Avodah Zarah 11b). And so wrote the Chid"a (Brit Olam 435), and in (Shirei Bracha, Yoreh Deah 142). So wrote Rav Haim Palagi (She'elot U'Teshuvot, Haim BeYad 26), and he wrote that it's a very BIG sin. So wrote Rav David Zilberstein (Sh'vilei David 154). So wrote Rabbi Ovadia Hadiah (Yaskhil Avdie, Helek 8-20:56). So ruled the Ram"a (Mapah, Orah Haim 156). Also written in She'elot U'Teshuvot Sha'ar Efraim 24. So wrote the Peri Megadim (Shulhan Aruch, Yoreh Deah's commentary 65:45), and see Shulhan Aruch Yoreh Deah 150:1).
Possibly this could be a problem due to "deriving benefit from avoda zara".
There is also a problem with getting up close to a house of a"z worship (See Masechet Avoda Zara 17a).