Well, there has been more than one comment posted in reply to your inquiry, so you may feel satisfied with the answers you have received, but none of them addressed something I have noticed about the Turkish language and that is this:
When a verb has an object, the object takes the accusative case when it is definite and when a noun is pluralized it seems to take on a definitive quality that I think I always have seen as declined with accusative case (at least here in duolingo).
Having said that, I conducted a little experiment. I did a Google search of Turkish pages using "ben hayvanlar seviyorum" and then again with "ben hayvanları seviyorum". Interestingly enough, results were produced for "ben hayvanlar seviyorum" but upon closer inspection, those pages actually used the construct with "hayvanlar" in the accusative -- hayvanları. And the construct "ben hayvanları seviyorum" returned 260 times the results of "ben hayvanlar seviyorum" -- 2,340:9.
So, while I don't know that Turks have any specific grammar rule that addresses this, if "ben hayvanları seviyorum" is an example, at the very least, thinking of pluralized objects as having a definitive quality appears to be a good rule of thumb.
Note: After posting this, I stumbled upon something I had seen earlier that also might be of help to you. The link to it is here: https://www.duolingo.com/comment/7736911. You may have already seen it and/or mastered this concept by now, but for those who haven't it may be useful.
The above sentence is not consistent with the rules give by the link https://forum.duolingo.com/comment/7736911. According to these rules, we should have: "Köpek hariç hayvanları seviyorum": I like the animals except for dogs "Köpek hariç hayvan seviyorum.": I like animals except for dogs Why is there in this case an other rule applying?
If I understand your question, GordonRobb, you're not asking about köpek. You were asking why hayvanları and not hayvanlar. Am I right? I up-voted your comment, because I had same question. I know that you do use whatever the heck case it is for pronouns with "sevmek" (seni seviyorum). But do other nouns need the suffix as well with sevmek?
Yes. However I think I know now. I need to remember that there's a subject and an object. And there's an indirect object and a direct object. And when there's a direct object, it's the one that gets the accusative. I also, need to remember to read the sentence in a more literally translated way. So this becomes "except for dogs, I like all animals". When I read it this way, I get why animals is the direct object, and therefore gets the accusative - I think :)