Disclaimer: The following is largely my opinion, though I try to be as objective as possible with regard to logical counters. As for my credentials, I have 370 Dominion games played with at present 196 wins. I hover between a 1.1 and 1.3 win ratio, playing primarily bot lane.
Traditional tier lists are pointless in Dominion. Tier lists are designed to show which champions have the most solo-queue carry potential or the fewest popular team-play counters. As carrying games is far less straightforward in Dominion, a solo-queue tier list has virtually no use: it is difficult to tell which player carried thanks to the obscure, largely irrelevant points system and the lessened importance of K/D/A. Similarly, Dominion roles are very loosely defined and thus picking and counter-picking is currently more based on eliminating annoyances than securing/banning serious threats. (For example, Rammus is a widely accepted pick-or-ban, yet backdoor Rammus rarely works against coordinated teams and he is far from the strongest defender/initiator.)
Frankly, the only aspect of Dominion champion selection that can be quantified right now is the matchup at Bottom lane. As Bot is mostly a 1v1 battle, the matchup can be won at champion selection if the teams pick and counter-pick accordingly. In many ways, this can mean the team that picks the counter-laner has a far better chance of winning the game, though let me stress that this is not always the case and many teams can recover from a mismatched Bot lane. Also be aware that just because a champion is a hard-counter does not mean they will automatically win the lane. Player skill comes into play, as well.
The current "top" Bot picks are Heimerdinger, Yorick, Soraka and Ryze*, so the list will begin with them. Note that from then onward, the list will be in no particular order. The only ordering-by-strength in this list will be the strength of the three champions that best act as counters to the named champion. These counter-picks are taken from my experiences as well as logical deduction. Some may be incorrect, and I encourage you to enlighten me to both mistakes and potentially harder counters.
(*Ryze is arguably better as a top lane champion, but I list him here anyway as he is a strong overall pick)
Heimerdinger-- Sivir, Cassiopeia, Morgana
The key to defeating Heimer is to out-push him while staying behind minions to avoid his harass.
Yorick-- Nasus, Sivir, Nunu
Yorick has great sustain, but needs to get in melee range to do any real damage.
Soraka--Kog'Maw, Karthus, Zilean
Great sustain and harass, though she doesn't have much range. Take advantage.
Ryze--Kog'Maw, Galio, Yorick
Push, push, push, and stay away from him!
Sivir--Kayle, Cassiopeia, Malzahar
Kill her. You almost certainly can't outpush her, so farm as needed but keep putting pressure on her. She likely won't have much magic resist, as she'll either want glass-cannon or Atmog's.
Morgana--Cassiopeia, Kog'Maw, Teemo*
Harass her and keep her at a distance. If you push to her tower, be mindful of her ult!
Kayle--Yorick, Nasus, Malzahar
She's squishy and lacks range. Lategame she will put out ridiculous damage, so take the early advantage and do your best to keep it!
Nasus--Sivir, Kog'Maw, Heimerdinger
Push and stay out of range. Don't worry about his Q farm, it won't get high. Be careful when he gets tanky!
Kog'Maw--Graves, Cassiopeia, Kayle
Be aggressive! He can poke you to death, and out-range you, but in a straight fight he doesn't have a chance. Push early, too--he can't outpush you until late in the game. Pay attention when he uses his W(Bio-Arcane Barrage)--when it's down, he'll be vulnerable for about 8 seconds.
Karthus--Sivir, Kayle, Morgana
Push as hard as you can, kill him if he gets close. He's slow, he can't kite you, but try not to engage him if you don't have your shield up.
Cassiopeia--Yorick, Graves, Kayle
She's squishy. Avoid her skillshots, but kill her!
Graves--Yorick, Nasus, Heimerdinger
He needs to get close to be really bursty. Exhaust him when he does and you have a free kill and cap. Otherwise, push hard and watch for smokescreen--it might be a setup for a gank!
Malzahar--Morgana, Singed*, Gangplank*
Watch out for his ultimate. You may want to get a Quicksilver Sash, just in case. Push hard.
Mao'kai--Malzahar, Brand*, Galio
Harass him hard, burst him down if he becomes aggressive.
Galio--Sivir, Graves, Talon*
Galio is a decent counter to any AP champion. However, he falls short against AD. Harass him early and force him to build armor, and you'll mess with his overall effectiveness.
Anivia*--Morgana, Ryze, Cassiopeia
Anivia pre-6 is a joke. Post-6 she can stop mostly any push and put out strong harass. Most importantly though, she is excellent at securing ganks and controlling the health relic thanks to her wall. You want to out-push her early on and put her on the defensive, then harass her when shegets her ultimate and force her to waste time fighting back. If you can kill her that's always a plus--but remember her egg!
Singed*--Yorick, Cassiopeia, Soraka
He isn't afraid of you. He'll run around trying to fling you into his poison. He *should* be afraid of you, though, as early on he won't be able ot kill you nor will he survive much of your harassment. Remember that he needs to run past your minions to push, and hit him whenever he comes around. (Note, this strategy subject to change. It's based on logic; I haven't faced enough Singeds to provide evidence from my experiences)
Nunu-- Cassiopeia, Kayle, (?; Input requested!)
Keep him away from units and don't get caught in his ultimate. If you can manage that, he's not too tanky and not too threatening--out-harass him!
Riven*-- Nasus, Kayle, Cassiopeia
Riven's strongest point is midgame. Early she is threatening but cannot push all that well; lategame she falls off hard as she either builds tanky and deals mediocre damage or builds damage and is far too squishy to deal with lategame-oriented champions like Nasus and Kayle. Keep consistent pressure on her and do not let her combo you to death before you have items. Note: For the first 10-15 minutes, Riven is a very strong laner, but she needs to be complimented by a lategame bot laner that can switch with her, or she'll be overwhelmed in the final moments of the game.
Note: This list is incomplete. If a champion is not listed here, it is because I have not faced them often enough to discern a counter, have not played them myself, or have not been shown evidence of their success as a bot laner.
*Denotes a champion that is better off going top, IMO, but works as a counter pick.
11/7/11--Added Ryze* as a counter to Mao'kai. Added Galio, Anivia* and Singed* to list of Bot-Laners.
11/9/11--Adjusted Ryze counters. Replaced Heimerdinger with Morgana after a particular match that evidenced how strong Black Shield and Soil are at shutting down Ryze.
11/11/11--Adjusted Ryze counters again. Removed Sivir and Morgana, replaced with Galio and Yorick respectively. Thanks, Nekrogen for proving me wrong!
Added Nunu to bottom laners. Counters pending. Added Riven to Nasus counters, temporarily next to Heimerdinger.
-Added Cassiopeia, Soraka to Singed counters
-Added Cassiopeia, Kayle to Nunu counters.
-Adjusted Yorick counters. Heimerdinger removed, replaced with Nunu.
-Added Riven to bot laners with a note attached.
-Adjusted Kog'Maw counters. Teemo removed, replaced with Kayle
-Removed Riven from Nasus counters after several matches where my Nasus destroyed Riven. Honestly though, I'm beginning to think Nasus may be a pick-or-ban as he becomes unstoppable lategame.
11/22/11--Added Gangplank* to Malzahar counters.