What are some good ways of identifying private entities that may be inclined to accept intern work or offer an entry level job?
Find Areas of Need in the Job Market:
I can't stress this enough: analyze the needs of the current job market. Watch sites like GIS Jobs Clearinghouse (http://www.gjc.org) to identify skill that are in demand. Make sure you're learning and practicing skills that make you employable.
Know Where to Look:
In my community most entry-level GIS roles are created by the utility companies, the city's planning department, and occasionally, the county (usually the Assessor's office). In addition, there are a couple established engineering firms that hire entry-level GIS grads. You may be able to land GIS work with a campus researcher, but I believe off-campus positions are more helpful on your resume for most opportunities. Anything is better than nothing, though. If you're really really lucky, your community is a hotbed of GIS activity with opportunities in user-support, consulting, research (government or non-profit), programming, et cetera.
Keep it Real:
Avoid spending too much energy and enthusiasm applying for positions through the job sites. When you do apply for posted jobs, realize that many people are chasing these same jobs, which means good odds you're competing with someone who's perfectly-qualified for the position. Even more importantly, your odds of getting an interview for a posted position decrease with increasing distance to the hiring organization (this calls on Tobler's first law of geography, by the way). While I was a GIS student, I wasted a lot of time applying for posted positions when ultimately, my first job was not advertised. In line with @blord-castillo's advice, I met my future boss at a GIS users meeting, two years before I ever applied to his company. Toward the end of my last year of school I contacted him with my resume and fortunately, he gave me a job (as a programmer). So put yourself in the middle of your local GIS community and start introducing yourself. Attend every gathering you can; don't expect one lone users meeting to promote you into that coveted entry-level position.
Separate Yourself from the Competition:
Enfin, I think the average GIS student graduates under-experienced with respect to a few core skills and areas of awareness. They include: 1) writing SQL commands, 2) familiarity with non-ESRI GIS tools, like GDAL/OGR, and 3) programming anything for GIS, particularly data-cleaning scripts and/or custom tools/utilities for desktop or web applications. If you want a DIY crash-course in these topics, check out Python Geospatial Development by Erik Westra; it's a good place to start.
That's just my two-cents on this one. :)
I would encourage students to think outside of the box when attempting to identify places to work or accept you for an internship. If you are in school and willing to work for little to no money, you have limitless opportunites.
Rather than just scouring all of the "go-to" GIS jobs websites (like everyone else looking for a job), I encourage you to:
You will undoubtedly get a number of rejections, including the obligatory "s/he's in a meeting" from the front desk. Just ask to get their email address, phone number and contact them directly later to set up an appointment (the front desk is the "gatekeeper" that will try to keep you from meeting with those people). Keep trying with multiple organizations and you just might land yourself a "dream job" because you are the one who created it, and you won't be competing against 100 other dull resumes trying to catch the HR director's attention.
Lastly, be willing to volunteer your time. In college I once turned down an unpaid summer internship to instead take minimum wage job to make a few bucks over the summer. Later I interviewed for my first job out of college and learned that I was their second choice... why? Because the other person had a college internship on their resume to show some "real-world work experience".
Local User Groups. Show up and introduce yourself as someone seeking entry level work. In person networking will always work the best. ESRI organizes local user groups in metro areas; otherwise Meetup is a good way to find GIS groups or groups tangentially related to GIS.
It sounds like you recently graduated. Have you talked to your professors? Many of them might (or at least they should) have industry connections, either directly with people they work with/for or through past students that they keep in touch with. I have many friends that are constantly leaning on past academic advisors for new hires. Advisory boards are another place to look - many academic departments are now establishing these to keep in touch with the industries and to help in placing students such as yourself. See if there is such a board in your academic department, if so, get introduced to some members by an academic advisor.