I think one of the best card games for three players is Skat (not to be confused with the Scat mentioned in another answer).
This is a pretty heavy game on the level of Bridge, so you can decide whether it's a good fit for your group.
I personally prefer Ninety-Nine which is like Oh Hell!
with a really funky bidding technique.
You might also find this list on pagat.com helpful.
It seems like the numbers outside square brackets are the recommended numbers of players.
boardgames.stackexchange.com/users/118/Don Kirkby
When I was at university we often had exactly this problem.
The way I solved it was by extensive googling for a card game that was designed for three players.
The game I ended up finding was 3-5-8 .
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It has an interesting imbalance that means you spend a lot of time deciding which of your opponents is actually in the lead, and you often get the choice to give a trick to one or the other.
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I think I actually came across 8-5-3 , where the order of the hands is reversed.
Very similar games are 9-5-2 and Sergeant Major .
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500
Rummy is the best game of the rummy family IMHO, and it works well with any number from 2-5. klabberjass is a trick taking game that works only with 3.
I'll admit upfront that I've never played, but I assume its a good 3 player game as no other amount can play.
psychological jujitsu is great with 3.
It's a game of skillful bluffing and psychological matching of wits.
Egyptian Ratscrew is really fun with anywhere from 2-6.
It's a game of speed,
so if you're group doesn't like moments where you're all racing to slap a pile of cards, then I wouldn't try it.
But try it at least once, it's a lot more fun than it sounds. :D
boardgames.stackexchange.com/users/191/Gordon Gustafson
Since you mentioned the games you enjoyed playing, I'll point out they all can be played three-handed, with minor rules modification.
I've enjoyed playing Hearts three-handed.
Hearts with three or five players Spades for Three Players
Three Player Buck Euchre Where I live
, the canonical three-player card game is Cribbage .
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Cassino A brilliant card game that just magically deals out evenly to both 2, 3 and 4 players.
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The easiest way to get a three person game is to take a two person game with an extra person.
Poker is the one that immediately comes to mind. .
It's also possible to have "three handed" gin rummy, with one dealer and two players in every game.
The loser of one game becomes the dealer of the next.
It's possible, but harder, to downgrade a four person game into three.
In a pinch, I've played "three handed bridge.
" Deal out all four hands, highest bidder gets dummy, score each person separately.
On the other hand, "go fish," often a four player game, can be downgraded to three relatively easily.
boardgames.stackexchange.com/users/1344/Tom Au
Durak (Fool) is a Russian game that plays very well with 3 people.
The mechanics are also quite distinctive; it's rare to find a good card game that isn't built on trick taking in my experience.
boardgames.stackexchange.com/users/1565/BenOfTomorrow
I'd recommend: 31 (or Scat...
but I've never heard it called it that)
http://www.pagat.com/draw/scat.html 3 handed Pinochle http://www.pagat.com/marriage/pin3.html or Oh Hell!
http://www.pagat.com/exact/ohhell.html 3 is pretty fun with the right group, and all of these work great.
boardgames.stackexchange.com/users/1273/mcw
Briscola is a very Euchre-like trick-taking game that can scale to many different player numbers.
I prefer it with 3 so it sounds like the right fit for you folks.
If your group knows Euchre, they'll be able to learn Briscola quite easily.
Traditionally, it's played with Italian playing card deck (cups, swords, clubs, coins), but can easily be played with a modified International deck (hearts, spades, diamonds, clubs) by removing the 8, 9, and 10 of each suit.
boardgames.stackexchange.com/users/1658/Manata
Canasta has a three-player variant, where the players all compete against each other instead of playing in teams of two.
boardgames.stackexchange.com/users/1723/Stuart Cook
We have a great time playing Hearts, Spades, and Euchre with 4 players.  Oftentimes though we have three people available to play but not a fourth.  How can I find card games that are interesting for 3 players?
Card Games for Three Players?
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boardgames.stackexchange.com/users/1640/Dave
Skat, Ninety-Nine and Rummy are all good card games for 3 players.  One of the best card games for three players is  Skat. There is also Ninety- Nine or psychological Jujitsu. A brilliant card game that deals out evenly to 2, 3 or 4 players is Cassino. Hearts can also be played with three. Three handed gin rummy can be played, with one dealer and two players in every game and poker can also be played with three players. 
 On the other hand, "go fish," can be downgraded to three easily. And Durak (Fool)  is a Russian game that 3 people can play. Briscola  is a very Euchre-like trick-taking game that can scale to many different player numbers, whilst Canasta  has a three-player variant, where the players all compete against each other instead of playing in teams of two.
One of the best card games for three players is  Skat. There is also Ninety- Nine or psychological Jujitsi/ A brilliant card game that deals out evenly to 2, 3 or 4 players is Cassino. Hearts can also be played with three. Three handed gin rummy can be played, with one dealer and two players in every game and poker can also be played with three players.  On the other hand, "go fish," can be downgraded to three easily. And Durak (Fool)  is a Russian game that 3 people can play. Briscola  is a very Euchre-like trick-taking game that can scale to many different player numbers, whilst Canasta  has a three-player variant, where the players all compete against each other instead of playing in teams of two.