I enjoy many JRPGs, but also get frustrated with typical JRPG gameplay. Random encounters (often at a very high rate), difficult to navigate dungeons, save points (and often too few of them), and tedious turn-based combat (not that all turn-based combat is tedious, but ATTACK ITEM MAGIC DEFEND gets tedious, especially when you're forced to engage in it every three seconds) all make me want to throw the controller. Add to that most JRPGs I've played seem to have at least one if not several puzzles that are impossible to solve without using a walkthrough (or playing through the game several times through, which I'm sorry, I am a full time working adult; I very seldom have time to play a 60-100 hour game through more than once).
Where I find JRPGs excel is in character and story design; graphically they are often pleasing without being obnoxiously over-the-top. The JRPGs I stick with are the ones where I like the story and characters enough to put up with endless random encounters. (At the same time, I disagree with the notion that JRPGs corner the market on strong stories and characters. The best RPG story and character wise I ever played (IMHO obviously) was Planescape: Torment, obviously a Western game.
My favorite JRPG series are Shadow Hearts and Suikoden. Both first of all have some incredible stories and characters. Secondly, both of them shake up typical JRPG gameplay at least a little, making it actually fun for me. SH has its Judgment Wheel which adds some fun tension to combat (and it is customizable enough to accommodate different styles of play) and has some good tactical decisions to make with setting up combos, etc.
Suikoden has more typical JRPG style combat but also throws in massive army tactical battles and one on one duels which shakes things up nicely, plus has a pretty neat magic system. Some of its installments offer different things you can do with party formations and combo attacks which adds a nice tactical element. (However, Suikoden Tactics, which is obviously entirely tactical, has terrible (very slow) gameplay.)
More "typical" JRPGs like Final Fantasy or Dragon Quest tend to be hit or miss with me. Liked FF9, hated FF7 and couldn't even bring myself to finish it. I'm currently playing DQ8 which I was enjoying for awhile, but it's started to get tedious for me. At least that game makes it easy to reduce random encounters and escape dungeons (and get back to them) easily.
I could probably declare myself a JRPG fan for good and for life if they did two things. First is to either eliminate random encounters and make monsters appear on screen, for you to avoid or attack as you choose (they could even still randomly spawn in places, but I want to see them and have the option of avoiding contact with them). The second is to eliminate save points and allow saving any time you aren't in combat. My reasons for this are very simple: I have a life. If I need to shut off the console and do something else, I don't want to have to wander around for 45 freaking minutes just because I can't find a save point. I would be fine with a "QuickSave" feature like we see on a lot of DS JRPGs that exits the game automatically and deletes itself on load, with permanent saves still only available at inns or certain points. The fact is I don't play JRPGs as often as I used to because I know just because of save points, I will have to dedicate at least an hour or two at playing and hoping I can find a save point at the right time. If I just want to toss something on for 45 mins between other things I have to do, I can't throw on a JRPG. Console technology has passed the point where save points are necessary, and why not make the game friendlier to older or otherwise time-crunched players? Good story games shouldn't be restricted to the "hard core" player who does nothing but play one game for several hours a day. Some of us just can't afford to do that.
And will probably play them more often than JRPGs, in part because of that very reason.
