Project Managers almost never have a direct reporting relationship with the team members assigned to their project, but remain accountable for the project outcome. What are some effective ways of influencing commitment and participation when team members may have committments and priorities that conflict with participation in a given project?
First thing is to learn how much of priority your project does have.
Second thing is to actually ignore that information and fight for it like it was top priority thing and life of all people in the world depended on that one. It's actually working more often than you'd expect as it's so very common that priorities are decided only basing on whoever yells louder.
Third thing is to escalate importance of the project to people who actually have power to decide, so they can repeat after you: "my priority is more important than yours."
Fourth thing is to make the project cool, for whatever reasons. People actually prefer to work on cool project than on a boring one so they will likely run an extra mile to be involved in some cool venture.
Having said all of that I will refer to something I refer to pretty often: build trust relationship with people and you won't be afraid of their involvement at all - they'd either get really involved in your project or they'd tell you the sad truth. Either way you know where you stand.
I applaud you on understanding that even when we lack authority, we need to take responsibility. It is a tough road for a PM to walk and full of challenges. It can be done.
I think Pawel's suggestions are solid. I would switch around the order a little and offer a bit of caution on one.
Trust first, trust last: This boils down to relationship. There are three key powers in the work place, Role, Expertise and Relationship (You see seven listed in many texts, but they can all be traced to these core three). You don't have role power, even if you did, you have to be very careful and judicious with it. Expertise power is very hard to pull off as a PM. Even if you started as an Engineer, you are not one now and it doesn't take long for the Engineers to think "he's out of touch."
You need to build relationships from the absolute get go. This will build trust. Then you can start influencing the project. This is a go slow to go fast technique. When I joined my last company, I spent the first 90 days doing nothing more than holding a weekly status meeting and interviewing people. I listened. Three months in I was put on a failing project and pulled it out of its nose dive. I did it by leveraging my relationships.
Careful Escalation: Escalation can be very powerful. You also don't have to just escalate up. I have a PM technique I call Pothole Project Management. The concept is if you hit a bad pothole everyday, you probably won't be much influence to get it fixed. If the Mayor of your town hits it, you can be certain it will get fixed very quickly. I wrote a complete blog on this if you want to read more.
This said, you have to be very careful when you escalate. The kid who runs into his parents every time his brother so much as tosses a pebble, will soon get a reputation as a tattle tale. These kids are the ones that get ditched after school and picked last for dodgeball. You need to develop regular reporting and communication channels. Then instead of escalating, you just report. You make them easy to read and people will read them. Then you have a ready made venue for highlighting issues.
I always try to put things in the open as much as possible. There can be a valid reason why 'other' work gets more priority than your project. Not every project gets a 10 out of 10 for strategic importance.
The best way is to address it with th team member and the functional manager. If it cannot be resolved you discuss it with your sponsor and/or SteerCo. When they are not inclined to do anything about it, you adjust your project plan accordingly and get it approved.
When you do get priority, there is always a chance someone is not as philosophically about it as you are. So make sure you follow it up very closely and report every breach of the new 'agreement' immediately.
I recommend building great working relationships with the line managers of your team, as well as with the team members themselves. That way, there is a better chance that the managers will try to support you whenever possible, rather than pulling resources from your project when a resourcing issue erupts elsewhere.
I have been in a meeting where a line manager said "I've no idea what is going on with X's project, so I'll pull Y from it for a couple of days to help Z out of a hole. If X has a problem with that he can come and talk to me, but by then Z's problem will be fixed and X can get Y back." If X had communicated more effectively and built a better relationship with the line manager, this would not have happened.
Building these relationships will help to demonstrate to your team members that they are important to you, and it shouldn't need much encouragement from you to get them on side and supporting your objectives, rather than allowing themselves to be distracted.
Not completly on-topic, but one unspoken strategy to get things done is: annoyance. I annoy people not doing what I think needs to be done for the project, in order to have them do the stuff just to get rid of me. And it's working everytime! Team members, providers, contractors, other employees of the company somehow involved in the project. If they protest, just say "it's really important for the project to succeed", and what could they possibly answer?
Phone them, mail them, fax them, meet them, remind them, let post-its on their desk, call them at home, whatever. It cannot fail. When it's more annoying for them to listen to you than to do what you need (which they have to do anyway sooner or later, as it's actually their job), they will do it as quickly as possible.
After several times, they will even do it before you begin your annoyance tactic, and you actually trained them!