I have played them, and there isn't much to utilize the UA with (I try to give every civ a chance). I never play on small islands but I don't play on Pangeae either. On every Fractal map I play the UA has some purpose, just not a whole deal important.
Other than India's UA, I can't think of another UA that's more lackluster (maybe Ottomans). I even like England's UA, if that sounds ironic, and the extra spy is nice.
I don't really need to land/attack right away, so I never found the most optimal use for it. I have utilized siege units before, but they always run the risk that the city might pop a tireme at some point. If it's not one hit KO it makes the catapult an easy target for the city.
It's very good for early exploration - my current map (Sweden) would have been tailor-made for Harald, with a large number of large islands, most initially unoccupied, and nearly all with coastal access. Fast embarked movement in the early game and the ability to move a scout/settler onto land and explore or settle in the same turn is very nice. It may well be more an illusion of utility than actual use (in this game I grabbed most choice sites as Sweden, and there were still ruins by 1515, so no one else seemed to be doing much exploration; then again you can probably say the same for the value of Polynesia's UA), but it makes early-game exploring that bit more fun - and fun is, to me, what Denmark's about.
America 14
Arabia 23
Aztecs 23
Babylon 24
Byzantium 23
Carthage 23
Celts 16
China 24
Denmark 10
Egypt 15
England 16
Ethiopia 20
France 20
Greece 22
Huns 10
Inca 25
Iroquois 14
Japan 18
Korea 21
Maya 27
Mongolia 14
Netherlands 17
Ottoman Empire 20
Persia 22
Roman Empire 20
Russia 18 -2
Siam 21
Songhai 15
Spain 5
Sweden 19
With the restriction on having experience with civs to vote on them, the downvote is particularly tricky - I've tended only to play civs that look interesting... Greece goes down as being by far the least interesting of the diplomatic civs, and a civ I've played. As I and others have noted, its UA is no longer useful in any case, but I'm not a fan of passive modifiers that don't take work to use and Greece doesn't give you anything for your influence you won't get with any other civ (unlike Siam). Still, Alex is a very strong AI and Companion Cavalry are one of my favourite units in the game (except when Alex is using them as an AI - of all the UUs, this seems to be the one the computer knows how to use to most brutal effect).
It's taken me a few games to get the hang of Sweden, who I felt deserved a chance because of the interesting synergy in their UA. I rarely have trouble making long-term friends - in my current game I spent much of the mid-game with five DoFs (think Babylon bonus for all GP generation - and yes, it does apply to Generals and Admirals, although not as far as I can work out to Prophets). After a bizarre form of diplomatic chaos where I lost track of nearly all relations (remaining friends only with Korea, and then I think only by the skin of my teeth) and ended up in three wars, I've managed to get my bonus back up to 40%. Probably this is partly attributable to 'unwise' play to maximise the effect of the UA, befriending hated enemies, but I can often negotiate those kinds of situations so I doubt I was overreaching more than I would have otherwise. The GPs I've been putting out have helped strengthen my alliance with or ally with nearly every CS Austria hasn't picked up - I have eight CS allies so far (and 9 projected votes of the 10 I'll need in the UN when it gets built - 8 CSes, and the liberated Songhai). I even built The Louvre specifically to gain CS allies.
When you put out so many excess GPs it's no 'waste' to ally with multiple CSes using the spares, and once you hit the right Patronage policy and have enough allies, you benefit from the feedback of getting GPs periodically from CSes and giving them back to a CS (and very often fulfilling a 'Generate GP X' quest into the bargain). I'm specialising my GP production so that I mostly produce scientists (who I don't trade), merchants and artists - and indeed the Swedish UA is very good for disposing of semi-"unwanted" GPs that you generate incidentally because you want to maximise culture production from specialists, for instance. Quests and gold alone can usually net me strong alliances with several CSes, but not with most of the ones remaining on the board. While it seems almost always a better investment to give Generals and Admirals to CSes than to keep them (though this calculation would be different if I had a map where Citadels were useful), not to mention Prophets you capture of an unwanted religion.
To top it off, the Haakepeliita is nothing exceptional but is somewhat interesting in its design, and the Carolean is probably the best "basic unit with promotion" UU in the game.
EDIT: Though there is one downside - I hate the fact that the Swedish UA is called "Nobel Prize". It's overly silly and the fact that Alfred Nobel happened to be
Swiss Swedish is surely not very characteristic of Sweden. ADDITIONAL EDIT: Not sure why I wrote Swiss...
. The free GS and extra GS make the game more adaptable to any strategy because you need good 
...at least for me.
