Sean, I'm not entirely sure where that difference to my way of thinking is coming from, but while playing around with it I also realise that I'm much more willing to say "in the summer" than "in the spring".
I want to say that the "the" option is preferable when you're talking about the season of a specific year, but I think it's rather just a dialectical difference and either is fine.
It works like "yesterday", "today" and "tomorrow" in relation to "day" in pretty much any European language, just on a larger timescale. (Okay, it's not as impressive in English, but consider German: "gestern", "heute", "morgen", vs. "Tag".) They are tendentially rather old words, so they might have shifted away from their related terms, or they're just short forms of more explicitly day- or year-related words.
Jövőre is probably the easiest to see. Jövő means "future" and is the present participle of jön, "to come". So basically "that what's coming". I'm not sure about the -re ending, though. It might imply that we're "moving (on)to the future".
Idén for "this year" is a bit harder to make out. It derives from idő, meaning "time", but in earlier Hungarian it also meant "year". The -n suffix indicates that we're "on the year", i.e. we're talking about the year we're currently in. The old ide- root here is a form that you can still see in nowadays possessive forms: "az ideje" - "his/her/its time".
Tavaly likely derives from távol, meaning "far away" or "that which has been removed".