Mikan Sakura from Gakuen Alice. She's eternally optimistic, and happy. She's the EXACT DEFINITION of the Pollyanna. Even after Mikan discovers her alices and all the horrible stuff that happens to her, she always cheers everyone up. Even Natsume and Luka love her, eventually. And Natsume Truly and deeply loves Mikan. But really, who CAN'T love Mikan?
Excel Excel, the ultimate oblivious Genki Girl, who will always believe that she is Il Palazzo's favorite minion no matter how violently he rebuffs he, until Il Palazzo shoots her at the end of episode 23. Although he has shot her before, including with antimateriel weapons, during this episode the Toon Physics are turned off.
Tohru Honda from Fruits Basket. In the anime, the other characters use her as a doormat, but she remains eternally perky and eventually wins them over. In the manga, on the other hand, while she acts the same, she's also shown as doing some serious repression.
Maria Taro Sekiutsu, a grindingly poor illegal immigrant who frequently expresses rose-tinted views of Japanese life, is an only somewhat less extreme example.
In Hayate the Combat Butler, we have the titular Hayate. He has gone through some seriously heartwrenching moments as a child including: being sold off by his parents for frivilous reasons, being ostracized in school for having such parents and people assuming he's the same way, leaving his first and best friend in a fight caused and not being able to apologize because they were seperated, and having to work various degrading odd jobs to support his parents' gambling habit. Perhaps because of this, he strives to be the best butler he can be and flat-out refuses to burden people with his problems, so he smiles; whenever someone asks him about his past, he always answers cheerfully and never complains.
Hayate has actually gotten mad at and expressed hatred for his parents a few times.
Sora, protagonist of Kaleido Star, has a will of steel, considering the hoops she's had to jump through to get to the top (both figuratively and literally).
Sora is actually a massive subversion. Despite being super sunny and plucky, she does have several breakdowns ( to the point of temporarily leaving the Stage in the second season), and later has to settle for a somewhat less rose-tinted view of the world. Considering she's a massive deconstruction of Purity Sue, it makes sense.
Milfeulle from Galaxy Angel is almost impossible to make depressed or sad for very long, even when starving to death, being completely broke, having people actively trying to kill her, being dead...
Kasumi Tendo from Ranma ½. Nothing catastrophic happens to her, but she does seem to find herself witnessing catastrophe on a regular basis. Unless you count being possessed by a demon as catastrophic...
Nodoka Saotome is farmoreawareofher situation, but refuses to let it get to her, making her optimism seem heroic next to the passivity of Kasumi (who lives in a big house, surrounded by family, not a care in the world.) At least, until she starts crying in her sleep... and reenacting a Seppuku ritual.
Bleach: Orihime Inoue is ultimately a subversion. She seems to be this in the first arcs, but she's later revealed to be a Stepford Smiler who hides her massive self-esteem and lonely issues behind her smile... and once she's taken out of her more-or-less safe Karakura environment, she's subjected to a Break the Cutie process. She arguably plays this straighter when she's back home, but it can also be seen as a coping mechanism: she does state that she's had a rough life, but is trying not to dwell on it anymore.
Rico from Gunslinger Girl fits this trope, being overly happy with her new life as a cyborg-assassin, although it can likely be attributed to her aggressive mental conditioning.
If Nunnally Lamperouge of Code Geass isn't this trope, I don't know what is.
Subverted in R2. Boy, does she take it hard upon learning what her brother did to create a better world for her...
Subverted in Sonic X by Cosmo, whose entire family was wiped out by the Metarex right in front of her and who has frequent feelings of self doubt, Survivor's Guilt, shame for her own hatred of the enemy and unworthiness.
Brook from One Piece is an interesting male example of this trope. One of the characters almost Lampshades this by asking why he's so cheerful even though his life has been awful. It's heavily implied that Brook has gone insane from all of his trouble, though.
Really, most of the Strawhat Pirates fit this. Luffy fit this partly before the timeskip, what with witnessing Sabo dying, while trying to become a pirate, and still not giving up on his dream of being the Pirate King. After the timeskip he fits this even more, since he saw his brother, Ace, getting killed right in front of him, and he still managed to push past it and have a positive outlook on life and keeps following his dream..
Lacus Clyne, from Gundam SEED is a borderline example. She manages to maintain her childish idealism in spite of everything that happens, including her father losing his office to a genocidal maniac and then being murdered by him, along with most of his supporters and nearly Lacus herself, the nuking of several colonies and just generally living through a Gundam show. She did cry when she told Kira about her dad and was very upset when her Body Double Meer took a fatal shot for her, but other than that, she rarely breaks down.
Keiichi Segawa from the X1999 manga. Lost his father in early childhood, and then his mother is killed in one of the earthquakes caused by the Dragons of Earth, yet he remains optimistic and kind until finally being phased out of the series.
Usagi of Sailor Moon isn't as much of a Pollyanna as fanfic writers would have you believe (where Usagi is a perky saint and can never be pried from her hold on a "good in everyone" mentality), but she does have an overwhelming belief in the power of love in the anime. The manga and live-action are much darker, though.
Elmer C. Albatross from Baccano!. The woobie factor is subverted in that most people find his constantly chipper, love-everyone attitude to be inappropriate and incredibly insensitive: "C'mon, let's laugh together!" is not what a teenage girl wants to hear just after nearly being killed by her fiance's murderer. Pretty much the only person that doesn't find him creepy is the equally creepy Huey Laforet.
We eventually learn that his Pollyanna nature is actually a front. Not that he feels sad about anything. Just the opposite: he's a sociopath. He doesn't care about anyone, but he wants everyone to smile just because.
Nana from Elfen Lied is clearlybroken by the time we meet her and continues to have horrifyingly terrible things happen to and around her for the entire story. She nonetheless somehow clings to a generally positive outlook that everything will be okay, especially as long as she can be with papa again. She doesn't. He explodes
He does survive in the manga, though.
Nao Kanzaki from Liar Game. Altough when things go real bad, she panics to the point of tears.
Negi of Mahou Sensei Negima!starts as one of these, but when he begins training under Evangeline, she specifically makes it a point to knock that out of him, because the world is not that nice, and he needs to realize that. As it turns out, she wasn't entirely wrong.
No matter how many times Shibuya Yuuri from Kyo Kara Maoh is betrayed, he is always willing to give that person a second chance. When a girl is jailed for posing as his daughter and trying to assassinate him, he smuggles her out and adopts her.
InuYasha: Within the first chapter of her introduction, 7-8 year old Rin is revealed to be an orphan who watched her entire family be slaughtered by bandits in front of her eyes (a trauma that left her mute and continuing to suffer nightmares), who is scrounging a living on the outskirts of her home village, being beaten by the villagers for stealing fish to survive... and then she's murdered by wolf-demons. After she joins Sesshoumaru's group, she continues to be a target for kidnapping and attempted murder. Despite everything that's happened to her, she remains a constantly cheerful, upbeat person to the end.
The eponymous character of Helen ESP, who doesn't let being rendered blind, deaf, and mute and her parents dying get her down. Though, this is after some backstory Character Development from being a very bitter girl.
Takeru from Snow White And Seven Dwarfs. While he does have his moments when he's upset, he's generally upbeat and cheerful despite being in a Crapsack World, making it a point to try and remain positive no matter what happens.