Regardless, as an employee I've learned to either ignore these things outright, or at least give the most artificially bland non-feedback possible. On many of the occasions in the past when I've given meaningful feedback... despite the "anonymity", that feedback has been repeated to me verbatim by a manager in later conversations. Often in a passive-aggressively angry or intimidating way.
It's a problem of population size. For company survey data to be all that meaningful, you have to AT LEAST segment the "anonymous" responses by location, group, etc. So even if the text doesn't include your individual name... it still narrows your identity down to a couple dozen people in a HUGE company, and probably just 2 or 3 people in a small company.
With that small of a population, it's really not that hard for managers to piece together who said what. My experience is that when management receives feedback they'd rather not hear, often they REALLY do want to piece together who said what. So whatever your noble intentions are in creating an "anonymous" survey system, I believe that human nature works against you on this. You're really just creating a honeypot trap to help root out "negative influencers".
Talk to your people. I know that a lot of employees roll their eyes at regular "1-on-1's", and a lot of managers don't like doing them either (so they're often not done very well). However, I just can't comprehend how anyone could consider themselves an effective manager without having those with every direct report on at least a bi-weekly basis, preferably weekly. If you're doing those right, then you'll have a strong relationship within your team even if everyone doesn't always see eye to eye.
Of course, 1-on-1's and manager-employee communication at the team level is a separate matter from COMPANY WIDE feedback. Upper management can (and usually does) have blind spots due to information getting blocked by poor managers in between them and staff. If you have a solution for bad managers, or for bad organizational structures that leave employees with either multiple managers or none at all, then you'd be a messiah. Generally, that's the last thing in the world that upper management really wants to tackle, though.