What positive writing exercises improve happiness? Writing seems like a perfect medium for expressing ones problems but can it also be used to improve happiness somehow? Maybe through some sort of creative writing exercises that focus on positive thinking?

Are there any positive writing exercise that can improve happiness, if even temporarily?
Writing therapy

There is quite a lot of research on writing therapy and expressive writing more generally. As you note, much of this research relates to writing about negative emotional experiences. There is some support for the positive mental health benefits in writing about traumatic events (e.g., see the meta-analysis by Smyth, 1998).
While the interventions are focused on writing about negative events, the intervention

Positive psychology and writing

I have read about a few specific positive psychology writing interventions, such as those mentioned here:


  
  Keep a gratitude diary: Take the time each day to write down three things that went  well and why. This causes psychological well-being
  levels to increase in a lasting way.
  Thank a mentor: Write a letter of thanks to someone to whom you owe a debt of  gratitude such as a teacher or grandparent. Then visit the
  person and read the letter to  them. People who do this are measurably
  happier for more than a month.
  Learn to forgive: Let go of anger and resentment by writing a letter of forgiveness to a person who has wronged you. Inability to forgive is associated with persistent rumination.
  


Toepfer and Walker (2009) discuss writing letters of gratitude. They cite several studies showing positive effect, stating that


  Expressive writing studies are plentiful and the once anemic domain of
  letter writing as  a vehicle for improving health has seen a recent
  surge of interest (King, 2001; Sheldon &amp;  Lyubomirsky, 2006a; Seligman
  et al., 2005; Lyubomirsky, Dickerhoof, Boehm, &amp; Sheldon,  2009). For
  example, VandeCreek, Janus, Pennebaker and Binau (2002) asked
  participants to  pray and write letters to God and found that both
  prayer and the letters increased insight  and positive emotion, more
  so than simple written descriptions, where a single letter to God  had
  the most impact. The authors explained that the act of praying or
  explaining to another  (in this case in a letter to God) was more
  conducive to personal insight and greater positive  emotional
  formulations about life events. In other words, writing a letter to
  God was found  to improve participant’s positive feelings about life
  events.  
  
  Watkins, Woodward, Stone and Kolts (2003) conducted a study
  that examined mood  changes as the result of various gratitude
  inductions, one of which was a letter writing  condition. Their
  findings revealed that writing a gratitude based letter produced a
  positive  affect increase compared to the other gratitude inductions
  (Watkins et al., 2003).


Burton and King found positive effects from getting participants to write about positive experiences. They also summarise some of the literature noting that


  research has begun to explore a variety of writing topics that
  might be associated with health beneﬁts that do not focus exclusively
  on negative experience. King and Miner (2000) found that writing only
  about the positive aspects of a traumatic experience was associated
  with the same health beneﬁts as writing about trauma. King (2001)
  found that individuals who wrote about their best possible future
  selves showed physical health beneﬁts as well as enhanced
  psychological well-being after writing.


Burton and King's instructions were:


  Think of the most wonderful experience or experiences in your life,
  happiest moments, ecstatic moments, moments of rapture, perhaps from
  being in love, or from listening to music, or suddenly ‘‘being hit’’
  by a book or painting or from some great creative moment. Choose one
  such experience or moment. Try to imagine yourself at that moment,
  including all the feelings and emotions associated with the
  experience. Now write about the experience in as much detail as
  possible trying to include the feelings, thoughts, and emotions that
  were present at the time. Please try your best to re-experience the
  emotions involved.


As for a mechanism of operation, Wing et al present the following ideas:


  Writing about nontraumatic events likely shares some effects with
  writing about traumatic events. Writing about any meaningful aspect of
  life may promote cognitive processing, encouraging the examination,
  understanding, and assimilation of emotions that might otherwise be
  left unscrutinized (Pennebaker, 2002; Pennebaker, Mayne, &amp; Francis,
  1997; Pennebaker &amp; Seagal, 1999). Writing about a meaningful topic may
  result in enhanced emotional regulation, related to perceptions of
  self-efficacy and control over emotional experiences (Greenberg,
  Wortman &amp; Stone, 1996; King, 2001, 2002; Lepore et al., 2002). Writing
  may afford the writer the opportunity to gain a sense of mastery over
  his or her emotions and to clearly identify priorities, preferred
  outcomes, and goals (King, 2001).


References


Burton, C. M., &amp; King, L. A. (2004). The health benefits of writing about intensely positive experiences. Journal of research in personality, 38(2), 150-163. PDF
King, L. A. (2001). The health beneﬁts of writing about life goals. Personality and Social Psychology Bulletin, 27(7), 798–807.
King, L. A., &amp; Miner, K. N. (2000). Writing about the perceived beneﬁts of traumatic events: Implications
for physical health. Personality and Social Psychology Bulletin, 26(2), 220–230.
Wright, J. (2002). Online counselling: Learning from writing therapy. British Journal of Guidance and Counselling, 30(3), 285-298. PDF
SMYTH, J.M.(1998).Written emotional expression: effect size, outcome types, and moderating variables.
Journal of Consulting and Clinical Psychology, 66(1), 174±184. PDF
Toepfer, S., &amp; Walker, K. (2009). Letters of gratitude: Improving well-being through expressive writing. Journal of Writing Research, 1(3), 181-198. PDF
Wing, J. F., Schutte, N. S., &amp; Byrne, B. (2006). The effect of positive writing on emotional intelligence and life satisfaction. Journal of clinical psychology, 62(10), 1291-1302.