It can be very frustrating if you ever have to queue 50 min. for your LFR invite because you are not a heal or tank (btw nice game design where you have to wait 50 min to play the game).
How to do it
However, if you want to go AFK and not miss your invite you can go to your class trainer (e.g. in Stormwind) and queue up for proving grounds. When you are in the proving grounds you're queue is listed as "paused". However, your wait time is still counting up. After coming back to the keyboard you have to leave it via hearthstone (I have not been able to leave it via right-click on dungeonfinder icon). After leaving you pretty much get an instant invite.
**Why is this working?**Proving grounds work similar to LFR, BG, Arena and Dungeon instances. It is bascially a "1-person dungeon finder group".