Currently my biggest gripes with the LFR loot system are:
1: Too much RNG in getting a piece. Sure, some people get pretty lucky in LFR, but others go weeks and weeks without getting a single upgrade.
2: You can be AFK the whole time and still have just as much of a chance to get loot as someone who was actively playing at their best.
3: Getting duplicate items. Personally, I've only gotten a couple pieces from LFR, and I've been doing all five for the past five weeks or so. This week the only piece I got were bracers I got the first week.
I know that a lot of other people aren't fans of the new LFR loot system, and just yesterday GC tweeted that they'd like to add a way for you to choose what spec you wanted to get loot for.
My suggestion is this:
1 - Boss dies.
2 - Each person has a chance to be selected for loot. (possibly the same chance it is now, maybe more)
3 - If randomly selected, a message pops up that says something like: "You have been chosen to have your pick from the spoils of the vanquished foe! You may choose one of the items below, or decline the loot instead for a sack of the foe's gold coins."
4 - A window with class-appropriate loot from the boss's loot table pops up. You have about a minute or a minute and a half to pick which piece you want (timer indicated on window). If you don't choose a piece, you don't get loot.
This way it takes away some of the RNG by letting you pick what piece you want if system chooses you, and if it's class-specific loot instead of spec-specific loot, that also takes care of the issue of getting loot for the role you want.
Also - if it is done this way, it will feel more like an actual 'kill and share loot' process, and you won't be able to just be completely afk and get loot.