We currently seem to have several questions will good and relatively highly voted answers, but without any accepted answers.
What can we do to encourage people to actually accept answers to questions?
If the user is new to Stack Exchange, or has indicated that one of the answers has worked for them and has yet to accept an answer days later, a prompt of some kind is probably in order.
In the case where the user is new to Stack Exchange -- For example if the user has never accepted a question (check their profile and reputation to see if they might be new-ish to this) -- then a comment on the user's question with a little explanation about accepting answers would be a good idea in my opinion.
An answer on Stack Overflow Meta suggested this as a helpful prompt if it looks like a user might not understand the accept feature. I think it's on target.
If a particular answer is helpful in solving your problem, mark it as "accepted" by clicking the little checkmark next to it. If your questions are receiving unhelpful answers, clarify what you're looking for or leave constructive comments on the answers explaining how they fall short.
We could also add
For more on accepting answers see How does accepting an answer work?
Also see some good discussion in Is it bad form to prompt new questioners to formally accept your answer?
In cases where the user has clearly indicated that one of the answers has worked well for them yet has not accepted it days later then a prompt may be in order. For example if the user says "thanks, this worked" (or something to that effect) in a comment on one of the answers, yet has still not accepted the answer days later, we could consider adding a prompt as a comment.
Otherwise, if the user has reputation or accept rate to show that they grok the accept feature, and how accept rate works, then it may be best not to prompt. In some cases, good as the answers may be, they do not solve the orignal poster's problem.
I think, that Alex is right. I asked a question here and got a very useful reply. I was very happy and tried to upvote (but coudn´t due to too little reputation) but I thanked the guy via a comment. But when I came back lately there was a reply that explained, that I could "accept" an answer if it solved my problem. I was unaware of that, but after learning that fact I happily accepted the answer. I´m new to SE and I still need to figure out, what is allowed here and what not. Mybe a lot of other folks feel the same and: Not all users are native speakers. We have to figure out the rules in a foreign language.
I would just - at least for the next 3 months - tell people what to do. Give friendly guidance and people will respond and - most of the times - do what you asked them to do. In marketing speak this is the "call to action" :-)
It seems to me that this is an inherit problem for all SE forums and should probably be solved at a higher level by SE. Perhaps people with questions marked as unanswered could be sent an email with a brief explanation about why selecting an answer to their question is important. They could at least be told that following up with a response and selecting that as the answer is helpful to the community. Perhaps they found an answer on Google and can share that with everyone else should none of the existing answers be "correct".
I agree, there is an abundance of questions (normally by new users) where relivent answers are supplied but the answer is never marked (you can take your pick of reasons!).
Could this not be done on a priviledge scale? like 3000+ REP you can accept an answer on someones behalf (after 2 weeks of no new answers, OP edits), or via the review system?
Just a thought, I'd like to see more ownus on accepting answers too, or at least allowing knowledgable contributors or moderators to accept on behalf of users that register, ask a question, get an answer, disappear...