Recipe for Success
1) Enter Champion Select.
2) Greet teammates so they know you are connected. (ex. "Hey everyone.")
3) Make ban requests/suggestions. (ex. "I find Shen particularly unmanageable," or "H8 Shen") Expect statements like this to be met by responses from teammates such as, "I'd prefer Shen not be banned so that I might play him if I am Top," or "I find him particularly manageable so if I am Top, we do not need to ban him." Ultimately, try not to make a big deal over bans. It is 3 champions.
4) Make your role preference known as follows: "Preferably not _____ (ex. Jungle)"
5) If you see that several others would not prefer a certain role, volunteer for that role. If someone volunteers for a role because you did not want it, thank him. Consider honoring him after the match.
6) Once players have volunteered for unwanted roles, players should make more specific preferences known (e.g. "I'm really on a streak as ADC") until roles are determined through players deferring to more insistent teammates.
7) Later-picking players should immediately start planning trades with higher pick players who are going to be playing Top, Mid, or ADC, to decrease the chance of those positions being countered. Late-picks offer statements like, "Who're your best mids? I own a lot of typical APCs, as well as Kha, Talon and Zed."

Benefits
If every player can play even 3 roles competently and confidently, it will be very rare for anyone to be forced to play one of his worst two roles.
If every player can play 4 roles well enough to be playing them in ranked, it will be extremely rare for any one to be forced into his worst role.
Everyone will play a role he excels at or is at least comfortable with, and an effective team composition will result.

Disclaimer
If you don't use this method of communication- or amiably and honorably say something like "filling [any role]"- you should expect the following:
- Having verbal conflicts with teammates.
- Being forced to play roles you don't want to.
- Forcing other players into roles they don't want to play.
- Particularly immature players trolling because they didn't get their preferred role.
- Effectively losing a significant number of games before they start.

This is how coordinated champion selection works, which goes a long way in Solo/Duo. Good luck with your method should you use a conflicting one, and enjoy your eternal Bronze rating.

They should make people have to read this before they start playing ranked and take a quiz over it, if they fail, ban them from ranked for a few days until they actually pass it.

brump.

Have to bump this.
Last night I joined a ranked solo queue. My teammates proceeded to type out their preferred roles as quickly as possible, calling everything but jungle.
I typed, slightly slower, my typical "Pref not jungle."

My team immediately berated me saying "respect calls bra." Meanwhile, I'm 2nd pick.
So I said "Aside from that I'm 2nd pick and am not required to be as flexible as later picks are, I respect "Anything but __" a hell of a lot more than "I'm ___". "
Luckily, they processed the sense in my statement rather quickly and worked things out between them so that everyone was happy.

Weird how not going by who types the fastest and calls one specific role is significantly more likely to satisfy everyone...

For another anecdote:
I had another queue in which our 3rd and 4th pick each called "Mid" when we got in. (3rd pick was selecting the round before 4th.) 4th pick proceeded to ask over and over again for Mid, instead of saying what else he did or did not want to play. 3rd pick proceeded to ignore him and then pick mid when it was his turn. Meanwhile 2nd had taken Singed.
When it got to 4th, it was agreed that he was responsible for jungling since he never stated anything but requesting Mid. He then explains that he can't jungle, and offers to duo top. Our generous Singed says "you can support me, but I need my farm." 4th pick locks Riven instead. Singed then volunteers to jungle to make up for Riven's idiocy.
Singed proceeded to be ineffective at jungling/ganking (as expected.) Riven proceeded to feed top. Mid also fed, by the way. We didn't do well bottom, but boy was this eye-opening. Singed honored. Riven reported. LP lost because Riven had never read this thread.