A REASONable and BENEFITial request. :)

Please update the game to show what is said with /say above your char's head as well as in the chat log.
Other chat types (/tell, /shout, etc.) should remain in chat log (to reduce clutter as well); only possibly with the exception that /tell-ing another player should somehow notify them of the tell.

It is my honest and humble opinion that currently (imo devs have done a great job of cleaning things up) the single biggest issue plaguing this game is the lack of in-game communication.
I have seen many posts on various forums about why that is, but not one has hit the nail on the head.

The biggest issue with FFXIV's social development is the fact that you can NOT see other players trying to talk to you unless you happen to be looking at your chat box at that exact moment. And not everyone thinks of running around you using /wave emotes until they get your attention, nor wants to.

Nobody wants to bother chatting with people when there is a 90% chance they will not get a reply!!

(I spend hours in game sometimes wishing other ppl would talk to me only to find out they tried, and I missed it because I was busy at a behest or something else. Nobody wants to sit and watch their chat box for 10 minutes every time they message someone to see if they get a reply)

I would like that but make it optional, so that people who don't like that can just not use the feature.

I -really- liked when SE added the little chime sound to XI's /tell system. The option to "hold" social chatter in a temporary portion of the chat log was also nice. These were subtle enough to do the job without making an on-screen mess.

To little speech bubbles over people's head: I'm not sure if I agree or disagree with this because XIV has a very.... realistic feel to it (within its own context of course). And I think that if text appeared over a characters head it would completely interfere with the game's atmosphere right now. That being said SE could find a way to work around it and make it aesthetically pleasing to fit the game.

What I do to avoid those guilty "missed tells/talks" is utilize the tabs and the option to have multiple chat boxes SE added a few updates back. I have a big chat log that is filled with all the mess of combat and synthesis information sitting on top of a 3-4 line smaller chat box that ONLY displays chat channels. This way even if I'm in intense combat I can still read what people in the Linkshell are saying without getting the two jumbled up.

I think the chime idea would be great. I never played FFXI (tried but didn't happen) so I never seen that feature before.
As far as it goes with chat "bubble" I would rather there be no 'bubble' per se, just transparent text floating above ppls heads that could be turned off or on.

I'm glad you mentioned the multiple chat boxes. I wasn't aware that was a possibility, it should definitely make things easier for me from now on. :)

I would like to look at the person while he or she is talking to me. I won't want to look down at a chatlog to wait for a reply. Yeah, FFXIV needs to add a chatlog so it can make conversation easier and make the game less of an archaic mess. Final Fantasy XI has and archaic chat system. That's because XI is an archaic game that was released in 2001 with a limited game engine and archaic systems that date even to today. XIV isn't forced to NOT have a chat bubble system and it really needs it.