When I stopped being a high-school teacher and returned to uni to do my PhD, I had exactly this problem when I started running tutorials for first-year maths. We tutors were given a set of problems and were instructed to present solutions to them at the board.
Even having been a schoolteacher, and having some skills in engaging students in whole-class presentations, it was like pulling teeth a lot of the time, and for many students it was boring. Indeed, for me it was boring. Also, I had the fear that some students were not actually understanding how to do these things themselves, since they were simply watching someone else do them.
(This fear is confirmed now as coordinator of a Maths Learning Centre, where I see people who have been in lectures and lecture-style-tutorials, who just don't know how to start, and who have fairly basic misunderstandings that no-one has yet noticed. Also, even though these students have seen a quite high-quality presentation already, they still needed to have it explained individually anyway.)
I decided after a semester I would ignore my instructions and do something different...
What I did in tutorials
I organised the students into groups of about five, and gave each group a white/black-board space to work on. I divided the front board in the room into two or three sections, brought in a rolling whiteboard with two sides, and gave them whiteboard markers to use on the windows.
Then I asked them to work through the example problems together on the board. I told them the reason was so that they had experience trying things with someone present to give advice, because everything looks easy when someone else is doing it. I also said that it was always ok to ask the person using the board a question at any point about what they were doing. I usually directed them to have a go at a particular question first, knowing that it would be most useful for them this week.
After that I walked around the room listening to what they were saying and giving advice. If all the groups were stuck on the same thing, I'd tell them to skip that one and I would do it at the board in front of the whole class later. Sometimes I would notice a serious misunderstanding in a lot of people and stop them there and then to do a whole-class session at the big board.
I'd usually wrap up the session with a whole-class part (maybe 10-15 mins) where I went through one thing they all struggled with, or where I asked for volunteers to tell me something they learned today.
How it turned out
I certainly had a much better time teaching these tutorials than the old style! I didn't have to wait around for people to arrive to start the class, because they could get going on problems straight away, and I had more time with the individual students. I was able to learn all their names because I talked to them individually. As to the students, they seemed to enjoy themselves in the tutes, and they did seem to get a lot out of it. In whole-class discussions, they were much more responsive than in classes where I hadn't done group-work.
In 2008 I had about a hundred students across all my tutorials, so I made sure I did evaluations with them to see how they perceived my teaching in this style of tute. I had above 95% agreement with all the measures, with 100% agreement for "This person shows enthusiasm for encouraging student learning", "This person shows concern for students" and "All things considered, how would you rate the effectiveness of this person as a university teacher." In the free-response comments, about a half of the students mentioned the group-work as the best aspect of my teaching.
It is important to mention that the remainder of the free response comments said that the best aspect were my clear explanations and the way I talked about the details of problem-solving when I gave examples. So you can't rely totally on the format to get good results -- you have to actually be good at explaining stuff and focus on the skills they need when you do!
Some did say that they would like more whole-class explanations of lecture content. This comment from previous semesters is one of the reasons I started doing a whole-class session at the end of every tutorial. However I still beleive that the groupwork is more important for this type of class. In the end it's the lecturer's job to teach the content clearly and you as a tutor can't be responsible for "fixing" that. Plus, with this format you can actually tell what is in the students' heads and so know where to pitch your explanations anwway.
[Note I gave a similar answer to this question, but this one includes more detail on why I chose to do it in the first place and the various ways I knew it worked.]