Pro tip: Request LinkedIn testimonials/recommendations from former colleagues and project partners. They will help you get hired when applying for a new job.
As I mentioned in episode 21 of the Career Hacking Podcast, testimonials on your LinkedIn profile will seriously help you stand out from the crowd. It's easier for recruiters that don't know you to put faith in you when others speak fondly of you.
Try to send messages to a dozen people if you have that many that are relevant and that you feel comfortable with. I sent 10 emails like this last year and 4 actually followed through.
Here is an example letter that you could adapt send:

Hello _____,
I hope you're doing well!
As I'm nearing graduation and starting to pursue job opportunities, I working to bolster my LinkedIn profile to catch the attention of recruiters. Would you mind writing a quick recommendation on my profile based on the ________ (specific group project, research job/duties, etc.) that we worked on together in the next two weeks? I certainly enjoyed working with (or for?) you and grew professionally through the experience.
I'd be happy to reciprocate with a similar recommendation on your profile if you'd like. Alternatively, if you would prefer, I can draft something for you to adapt and post on my page instead.

Thank you in advance,
Ross

 

Here is a similar letter I sent to colleagues when I left my job at GE Healthcare:

Good Afternoon _______,
As my time on the PET team draws to a close, I have begun to put my affairs in order in preparation for the next chapter. Not knowing what exactly is in store for the future, I am updating my resume and jotting down notes about my experiences in PET for potential opportunities down the road. While it’s still fresh on our minds, would you please post a quick recommendation of me for my LinkedIn profile?  … even if your recommendation is for them to “stay far, far away from this candidate.” :)

If you would like, I would be more than happy to reciprocate and share something on your page as well.

It, of course, has been a pleasure working with you over the past few years, and I will remember you and the team very fondly.

Best Regards,
Ross

As a connection of mine on LinkedIn pointed out, you can boost the likelihood that you will get a response by feeding them a few ideas of what they could recommend when you reach. Otherwise, it could be challenging for the recommender to get started with a blank slate. You want to make it as easy as possible for them without actually writing the review for them. 

Crafting the Perfect LinkedIn Profile

Coaching