I work for a heavily linux based company headquartered out of the Philadelphia area, with our IT operations based in Harrisburg.
I must agree that just hiring someone out of college with no experience is a very risky thing, and has screwed us over a few times. Of course, the people right out of college get hired at $35K/year and lead admins make $60-$80K/year easily, but you have to ask yourself if it is worth that new guy calling you 10 times a night after 2 months of hands on training to ask you questions they should know by now, or is it better to just pay someone more who has experience?
Both methods can work out well, you just need to screen the entry level people extremely hard to make sure they can self manage themselves and provide them good documentation so they're not calling you at 2AM asking you the IP address for something stupid. Reward entry level people who excel by giving them very nice raises after a year of solid commitment.
For the OPs question, the best thing you can do if you do not have experience in an area that a company is looking for is to admit that upfront, but then tell them something else you have experience with that is close to that. If you're doing consulting now, setup as many example services (with something meaningful running on them) on your home network. Key things to focus on would be linux based software firewalls/routers, mysql/postgresql deployments, web based front-ends to easily retrieve information and make reports from your databases (there's several really good opensource solutions for this), apache webserver serving up some PHP, Apache Tomcat serving up some java, KVM or VMWare virtual machines, etc. Anything that you can show off to prove you have some experience with it is good (especially if consulting).
From there, get references from every job you complete to build up your portfolio. You can concentrate in several areas all at once if you are up to it: Sysadmin (as you mentioned), Network admin (managing linux based switches/firewalls/routers/etc), DB admin (obvious), and developer (shell scripts are more on sysadmin side of things; this would be more for heavy development that could be deployed on an enterprise level where you are allowed approximately 2 to 4 hours of scheduled downtime per year).
Also look into learning HIPPA healthcare laws (or similar laws if outside the USA). You'd be surprised how many healthcare related IT jobs there are out there that require *NIX experience. Also, from my personal experience, *a lot* of the current admins for many healthcare companies are utter complete shite (some are very good, but most are horrible), which makes it easy for someone with a good skill set to get in the door and advance quickly (although it may upset incompetent coworkers if you constantly make them look like fools). The company I currently work for contracts with hospitals and such, and you would not believe how many lead admins for hospitals that we have had to hand hold walk through the basic task of setting up a Site-to-Site IPSEC tunnel on the devices on their end (sometimes on devices that we have never personally worked with, but STILL need to walk them through it and appear to know more about the device than they do..).
I figured I'd throw this reply in here instead of burning all my mod points on this thread. :)